https://optimization.cbe.cornell.edu/api.php?action=feedcontributions&user=Dasogil&feedformat=atomCornell University Computational Optimization Open Textbook - Optimization Wiki - User contributions [en]2024-03-28T16:34:26ZUser contributionsMediaWiki 1.40.1https://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5452Exponential transformation2021-12-15T01:14:13Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
A posynomial begins with <math> x_1,...,x_n </math> where <math> x_n<br />
</math> are real non negative variables. <br />
<br />
variables.<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation where <math> x_n </math> is replaced with the natural logarithm base exponential <math> e^u_i </math> <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function after substitution is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Properties of exponent can be used to further simplify the transformation above resulting in the sum of the exponents with a natural logarithm base. <br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{{u_1}{a_{1k}}}+{{u_2}{a_{2k}}}....+{{u_n}{a_{nk}}}}} </math><br />
<br />
This simplification can be applied in any instance where the product of logarithms with the same base is present to simplify the transformed function. <br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 & i = 1,....,m \\<br />
& \quad g_i(x) = 1 & i = 1,....,p <br />
\end{align}</math><br />
<br />
Where <math> f_0(x) </math> is a posynomial function, <math> f_i(x) </math> is a posynomial function and <math> g_i(x) </math> is a monomial function. <br />
<br />
In this definition monomials differ from the usual algebraic definition where the exponents must be nonnegative integers. For this application exponents can be any positive number inclusive of fractions and negative exponents. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} </math> is substituted by y. <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties and combining the exponents with like bases as a sum:<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + {x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional simplification through properties of exponents and combining the products of exponential terms as the sum of exponents with the same base:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}+{8{u_2}}}} + 2{e^{u_1}}+{e^{{2u_2}+{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
Overall exponential transformations can be applied anywhere a geometric programming approach is taken to optimize the solution space. Some Applications may perform a logarithmic transformation instead of an exponential transformation. <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5449Exponential transformation2021-12-15T01:09:22Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
A posynomial begins with <math> x_1,...,x_n </math> where <math> x_n<br />
</math> are real non negative variables. <br />
<br />
variables.<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation where <math> x_n </math> is replaced with the natural logarithm base exponential <math> e^u_i </math> <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function after substitution is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Properties of exponent can be used to further simplify the transformation above resulting in the sum of the exponents with a natural logarithm base. <br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{{u_1}{a_{1k}}}+{{u_2}{a_{2k}}}....+{{u_n}{a_{nk}}}}} </math><br />
<br />
This simplification can be applied in any instance where the product of logarithms with the same base is present to simplify the transformed function. <br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 & i = 1,....,m \\<br />
& \quad g_i(x) = 1 & i = 1,....,p <br />
\end{align}</math><br />
<br />
Where <math> f_0(x) </math> is a posynomial function, <math> f_i(x) </math> is a posynomial function and <math> g_i(x) </math> is a monomial function. <br />
<br />
In this definition monomials differ from the usual algebraic definition where the exponents must be nonnegative integers. For this application exponents can be any positive number inclusive of fractions and negative exponents. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + {x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional simplification through properties of exponents and combining the products of exponential terms as the sum of exponents with the same base:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}+{8{u_2}}}} + 2{e^{u_1}}+{e^{{2u_2}+{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
Overall exponential transformations can be applied anywhere a geometric programming approach is taken to optimize the solution space. Some Applications may perform a logarithmic transformation instead of an exponential transformation. <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5444Exponential transformation2021-12-15T01:01:17Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
A posynomial begins with <math> x_1,...,x_n </math> where <math> x_n<br />
</math> are real non negative variables. <br />
<br />
variables.<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation where <math> x_n </math> is replaced with the natural logarithm base exponential <math> e^u_i </math> <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function after substitution is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Properties of exponent can be used to further simplify the transformation above resulting in the sum of the exponents with a natural logarithm base. <br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{{u_1}{a_{1k}}}+{{u_2}{a_{2k}}}....+{{u_n}{a_{nk}}}}} </math><br />
<br />
This simplification can be applied in any instance where the product of logarithms with the same base is present to simplify the transformed function. <br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 & i = 1,....,m \\<br />
& \quad g_i(x) = 1 & i = 1,....,p <br />
\end{align}</math><br />
<br />
Where <math> f_0(x) </math> is a posynomial function, <math> f_i(x) </math> is a posynomial function and <math> g_i(x) </math> is a monomial function. <br />
<br />
In this definition monomials differ from the usual algebraic definition where the exponents must be nonnegative integers. For this application exponents can be any positive number inclusive of fractions and negative exponents. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
Overall exponential transformations can be applied anywhere a geometric programming approach is taken to optimize the solution space. Some Applications may perform a logarithmic transformation instead of an exponential transformation. <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5443Exponential transformation2021-12-15T00:58:01Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
A posynomial begins with <math> x_1,...,x_n </math>where <math> x_n </math> are real non negative variables. <br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation where <math> x_n </math> is replaced with the natural logarithm base exponential <math> e^u_i </math> <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref> where u corresponds to the n variable for each instance<br />
<br />
The transformed function after substitution is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Properties of exponent can be used to further simplify the transformation above resulting in the sum of the exponents with a natural logarithm base. <br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{{u_1}{a_{1k}}} + {{u_2}{a_{2k}}}+....+{{u_n}{a_{nk}}}}} </math><br />
<br />
This simplification can be applied in any instance where the product of logarithms with the same base is present to simplify the transformed function. <br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 & i = 1,....,m \\<br />
& \quad g_i(x) = 1 & i = 1,....,p <br />
\end{align}</math><br />
<br />
Where <math> f_0(x) </math> is a posynomial function, <math> f_i(x) </math> is a posynomial function and <math> g_i(x) </math> is a monomial function. <br />
<br />
In this definition monomials differ from the usual algebraic definition where the exponents must be nonnegative integers. For this application exponents can be any positive number inclusive of fractions and negative exponents. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
Overall exponential transformations can be applied anywhere a geometric programming approach is taken to optimize the solution space. Some Applications may perform a logarithmic transformation instead of an exponential transformation. <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5441Exponential transformation2021-12-15T00:44:13Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
A posynomial begins with <math> x_1,...,x_n </math> are n real non negative variables. <br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation where <math> x_n </math> is replaced with <math> e^u_i </math> <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function after substitution is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 & i = 1,....,m \\<br />
& \quad g_i(x) = 1 & i = 1,....,p <br />
\end{align}</math><br />
<br />
Where <math> f_0(x) </math> is a posynomial function, <math> f_i(x) </math> is a posynomial function and <math> g_i(x) </math> is a monomial function. <br />
<br />
In this definition monomials differ from the usual algebraic definition where the exponents must be nonnegative integers. For this application exponents can be any positive number inclusive of fractions and negative exponents. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> <br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
Overall exponential transformations can be applied anywhere a geometric programming approach is taken to optimize the solution space. Some Applications may perform a logarithmic transformation instead of an exponential transformation. <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5435Exponential transformation2021-12-15T00:33:00Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 & i = 1,....,m \\<br />
& \quad g_i(x) = 1 & i = 1,....,p <br />
\end{align}</math><br />
<br />
Where f_0 is a posynomial function, f_i(x) is a posynomial function and g_i(x) is a monomial function. <br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
Overall exponential transformations can be applied anywhere a geometric programming approach is taken to optimize the solution space. Some Applications may perform a logarithmic transformation instead of an exponential transformation. <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5301Exponential transformation2021-12-14T04:02:21Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leq 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 i = 1,....,m \\<br />
& \quad g_i(x) = 1 i = 1,....,p <br />
\end{align}</math><br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> {y^n} is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
<br />
<br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
<br />
<br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5300Exponential transformation2021-12-14T04:00:16Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation applied to geometric programs.<br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leg 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 i = 1,....,m \\<br />
& \quad g_i(x) = 1 i = 1,....,p <br />
\end{align}</math><br />
<br />
===Exponential Transformation in Computational Optimization===<br />
<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through exponential transformation the constraints of a geometric program are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> y^n & is substituted by y </math> <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
<br />
<br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
<br />
<br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5296Exponential transformation2021-12-14T03:54:13Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints nonconvex optimization problems. The constraints for geometric programs are posynomial functions which are characterized by being positive polynomials. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> This is done through a simple substitution of continuous variables with a natural exponent and simplifying binary variables by removing the exponents due to the binary nature of the variable. The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Using exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize conventional NLP/MINLP solvers. This can be seen used in various real world optimization problem applications to simplify the solution space as real world applications have extensive quantities of constraints and variables. <br />
<br />
== Theory & Methodology ==<br />
Exponential transformation is an algebraic transformation <br />
<br />
In non linear programming an exponential transformation are used for geometric programs which are composed of posynomial functions in the optimization function and constraints. <br />
<br />
A posynomial function which can also be referred to as a posynomial is defined as a positive polynomial function. <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref><br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem where the objective function is a posynomial which is minimized. For Geometric programs in standard form the objective must be a posynomial and the constraints must be posynomials <math> \leg 1 </math> or monomials equal to 1.<br />
<br />
Geometric Programs in standard form is represented by:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 i = 1,....,m \\<br />
& \quad g_i(x) = 1 i = 1,....,p <br />
\end{align}</math><br />
<br />
===Exponential Transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria for a geometric program. In a geometric program the approach to solving efficiently is taken by transforming the optimization problem into a convex nonlinear problem. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. Through this transformation the constraints are also convex.<br />
<br />
In a special case binary exponential transformation can also be applied where binary variables are linearized since their possible allocation is 0 or 1. Binary exponential transformation can be done by using the following replacement <math> y^n is substituted by y </math> REFERENCE TEXTBOOK<br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> this creates a convex under estimator approach to the problem.<br />
<br />
Also as presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
Additionally simplifying further for binary variables by substituting <math> {y_1}^2 with {y_1} and {y_2}^2 with {y_2} </math> since <math> {y_2} </math> is either 0 or 1 and any exponents on the variable will not change the solution space:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
{\partial^2 f \over \partial {x_1^2}} & {\partial^2 f \over \partial x_1\partial x_2} \\<br />
{\partial^2 f \over \partial x_2\partial x_1} & {\partial^2 f \over \partial {x_2^2}}<br />
\end{bmatrix}<br />
</math><br />
<br />
<math>\begin{align}<br />
{\partial Z(u) \over \partial u_1} & = 10{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} & \quad{\partial^2 Z(u) \over \partial u_1^2} = 20{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} \\<br />
{\partial Z(u) \over \partial u_2} & = 40{e^{2{u_1}}}{e^{8{u_2}}} + 4{e^{u_1}}{e^{2u_2}} + 3{e^{{3}{u_2}}} & \quad{\partial^2 Z(u) \over \partial u_2^2} = 320{e^{2{u_1}}}{e^{8{u_2}}} + 8{e^{u_1}}{e^{2u_2}} + 9{e^{{3}{u_2}}} \\<br />
\end{align}</math><br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Many of these applications use exponential transformation to convexify their problem space. Due to the closeness with logarithmic transformation, usually a combination of the approaches is used in practical solutions. <br />
<br />
=== Mechanical Engineering Applications === <br />
In the paper “Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption” a global optimization approach is explored for the synthesis of heat exchanger networks. As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
<br />
<br />
=== Electrical Engineering Application: ===<br />
Applications for VLSI circuit performance optimization. In this application a special geometric program defined as unary geometric programs is presented. The unary geometric program is a posynomial as defined in the [[#Theory & Methodology|Theory & Methodology]] section. The unary geometric program is derived through a greedy algorithm which implements a logarithmic transformation within lemma 5. <ref> http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf </ref> While this is not a specific exponential transformation example logarithmic transformations are within the same family and can also be used to convexify geometric programs. <br />
<br />
<br />
=== Machining Economics: ===<br />
Applications in economics can be seen through geometric programming approaches. Examples and applications include to analyze the life of cutting tools in machining. In this approach they use exponential transformations to convexify the problem.<br />
<ref> https://link.springer.com/content/pdf/10.1007/BF02591746.pdf </ref><br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5252Exponential transformation2021-12-14T01:19:08Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints (posynomial) nonconvex optimization problems. Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. Through the use of exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize normal NLP/MINLP solvers. <br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem. Geometric Programs take the form of:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 i = 1,....,m \\<br />
& \quad g_i(x) = 1 i = 1,....,p <br />
\end{align}</math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
===Exponential Transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> Also presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {e^{{\frac{2}{3}}{u_2}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + {x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Due to the closeness with logarithmic transformation, usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Economics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5251Exponential transformation2021-12-14T01:18:05Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints (posynomial) nonconvex optimization problems. Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. Through the use of exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize normal NLP/MINLP solvers. <br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem. Geometric Programs take the form of:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 i = 1,....,m \\<br />
& \quad g_i(x) = 1 i = 1,....,p <br />
\end{align}</math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
===Exponential Transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> Also presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{\frac{2}{3}}^{u_2}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + {x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Due to the closeness with logarithmic transformation, usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Economics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5250Exponential transformation2021-12-14T01:16:31Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. In computational optimization, exponential transformations are used for convexification of geometric programming constraints (posynomial) nonconvex optimization problems. Exponential transformation creates a convex function without changing the decision space of the problem. <ref name =":0"> D. Li and M. P. Biswal, [https://doi.org/10.1023/A:1021708412776 "Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method]," ''Journal of Optimization Theory and Applications'', vol. 99, pp. 183–199, 1998.</ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. Through the use of exponential transformations, the overall time to solve a Nonlinear Programming (NLP) or a Mixed Integer Nonlinear Programming (MINLP) problem is reduced and simplifies the solution space enough to utilize normal NLP/MINLP solvers. <br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem. Geometric Programs take the form of:<br />
<br />
<math>\begin{align} <br />
\min & \quad f_0(x) \\<br />
s.t. & \quad f_i(x) \leq 1 i = 1,....,m \\<br />
& \quad g_i(x) = 1 i = 1,....,p <br />
\end{align}</math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form <ref name =":1"> S. Boyd, S. J. Kim, and L. Vandenberghe ''et al.'', [https://doi.org/10.1007/s11081-007-9001-7, "A tutorial on geometric programming]," ''Optimization and Engineering'', vol. 8, article 67, 2007.</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> I. E. Grossmann, [https://doi.org/10.1023/A:1021039126272, "Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques]," ''Optimization and Engineering'', vol. 3, pp. 227–252, 2002. </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
===Exponential Transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meets the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above, all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally, as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized, it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref name =":1"></ref> Also presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation. <ref name=":0"></ref><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification Application in MINLP ===<br />
<br />
The following MINLP problem can take a convexification approach using exponential transformation:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{x_1^2}{x_2^8} + 2{x_1} + {x_2^3} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {x_1} \leq 7{x_2^{0.2}} \\<br />
& \quad 2{x_1^3} - y_1^2 \leq 1 \\<br />
& \quad x_1 \geq 0 \\ <br />
& \quad x_2 \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align} </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2 </math> by substituting <math> x_1 = e^{u_1}</math> and <math> x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad {e^{u_1}} \leq 7{e^{0.2{u_2}}} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad e^{u_1} \geq 0 \\<br />
& \quad e^{u_2} \leq 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
<math>\begin{align}<br />
\min & \quad Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {e^{{3}{u_2}}} + 5{y_1} + 2 {y_2^2} \\<br />
s.t. & \quad u_1 \leq \ln 7 + 0.2{u_2} \\<br />
& \quad 2{e^{3{u_1}}} - y_1^2 \leq 1 \\<br />
& \quad {u_2} \leq \ln 4 \\<br />
& \quad y_1 = 0,1 \quad y_2 = 0,1<br />
\end{align}</math><br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be shown to be convex through the positive-definite test of the Hessian. For the example above, the Hessian is as follows <ref> M. Chiang, [https://www.princeton.edu/~chiangm/gp.pdf "Geometric Programming for Communication Systems]," 2005. </ref>:<br />
<br />
In order to prove the convexity of the transformed functions, the positive definite test of Hessian is used as defined in ''Optimization of Chemical Processes'' <ref> T. F. Edgar, D. M. Himmelblau, and L. S. Lasdon, ''Optimization of Chemical Processes'', McGraw-Hill, 2001.</ref>. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above, the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore, H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently, various applications of exponential transformation can be seen in published journal articles and industry practices. Due to the closeness with logarithmic transformation, usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund, they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>K. J. Björk and T. Westerlund, [https://doi.org/10.1016/S0098-1354(02)00129-1, "Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption]," ''Computers & Chemical Engineering'', vol. 26, issue 11, pp. 1581-1593, 2002.</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Economics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described within [[#Theory & Methodology|Theory & Methodology]] are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5220Exponential transformation2021-12-13T03:29:46Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve. Many current applications use exponential transformation to simplify solution spaces and allow for solving with normal NLP/MINLP solvers. <br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5218Exponential transformation2021-12-13T03:26:52Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
<br />
Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve. Many current applications <br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {{e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5217Exponential transformation2021-12-13T03:25:34Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
<br />
Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve. Many current applications <br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{e^{{3}{u_1}}}*{e^{{-4}{u_2}}} + {{e^{{2}{u_1}}} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + {\frac{x_2^3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5216Exponential transformation2021-12-13T03:15:19Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
Exponential transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> The transformation can then be verified using the Hessian positive definite test to confirm that it is now a convex function. <br />
<br />
<br />
Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve. Many current applications <br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + {\frac{x_2^3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5215Exponential transformation2021-12-13T03:10:27Z<p>Dasogil: </p>
<hr />
<div>===Example of Convexification in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
<math> min Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
<math> s.t </math><br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex.</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5205Exponential transformation2021-12-13T02:10:57Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformations of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve approach.<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5204Exponential transformation2021-12-13T02:06:53Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve.<br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + \frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5202Exponential transformation2021-12-13T02:04:43Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve.<br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + \frac{x_2^3} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5201Exponential transformation2021-12-13T02:02:57Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve.<br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + {\frac{x_2^3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5199Exponential transformation2021-12-13T00:49:53Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve.<br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs<br />
<br />
A geometric program is a mathematical optimization problem <br />
<br />
Geometric Programs take the form of:<br />
<br />
<math> min f_0(x) </math><br />
<math> s.t. f_i(x) \leq 1 i = 1,....,m </math><br />
<math> s.t. g_i(x) = 1 i = 1,....,p </math><br />
<br />
convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf <br />
<br />
this would apply to any polynomial. <br />
A posynomial is defined as define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
===Exponential transformation in Computational Optimization===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
===Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
===Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1} + {\frac{x_2^3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1} and x_2 = e^{u_2} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
In order to prove the convexity of the transformed functions the positive definite test of Hessian is used as defined in "Optimization of Chemical Processes" <ref> T.F. Edgar, D.M. Himmelblau, L.S. Lasdon, Optimization of Chemical Processes. McGraw-Hill, 2001.</ref> can be used. This tests the Hessian defined as:<br />
<br />
<math> H(x) = H = \nabla^2f(x)</math><br />
<br />
to test that <br />
<br />
<math> Q(x)\geq 0 </math><br />
<br />
where <br />
<br />
<math> Q(x) = x^THx </math><br />
<br />
for all <math> x \neq 0 </math><br />
<br />
for functions the Hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
In the example above the hessian is defined as:<br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
Therefore H(x) is positive-definite and strictly convex. <br />
<br />
==Applications==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
*As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
==Conclusion==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
==References==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5167Exponential transformation2021-12-12T03:57:11Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve.<br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs <br />
<br />
define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
=== Exponential transformation in Computational Optimization ===<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
=== Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
=== Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} and x_3 = e^{u_3} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 + 4{u_3} \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> {u_3} \geq 0 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5166Exponential transformation2021-12-12T03:56:08Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
Information regarding the algebraic properties of exponential transformation <br />
<br />
In computational optimization exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> Through the use of exponential transformation the time to solve an Non-linear program (NLP) or a Mixed integer non-linear program (MINLP) is reduced by allowing the use of a global solve.<br />
<br />
<br />
<br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation can be applied to geometric programs <br />
<br />
define polynomials: <br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
<br />
= Exponential transformation in Computational Optimization =<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
<br />
=== Simple Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
=== Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation to continuous variables <math>x_1, x_2, x_3 </math> by substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} and x_3 = e^{u_3} </math> described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification through properties of natural logarithm:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 + 4{u_3} \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> {u_3} \geq 0 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. (Add jump metric) Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=5165Exponential transformation2021-12-12T03:23:59Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
=== Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
=== Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 + 4{u_3} \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> {u_3} \geq 0 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3717Exponential transformation2021-11-27T22:01:17Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
(eq 1)<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
=== Numerical Example ===<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
=== Example of Convexification application in MINLP ===<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 + 4{u_3} \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> {u_3} \geq 0 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3714Exponential transformation2021-11-27T21:58:14Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
(eq 1)<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_{1k}}{x_2}^{a_{2k}}....{x_n}^{a_{nk}} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_{1k}}}{e}^{{u_2}{a_{2k}}}....{e}^{{u_n}{a_{nk}}}} </math><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 + 4{u_3} \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> {u_3} \geq 0 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3708Exponential transformation2021-11-27T21:47:20Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
(eq 1)<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \leq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \leq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> u_1 + 4{u_3} \leq \ln 7 + 0.2{u_2} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> {u_2} \leq \ln 4 </math><br />
<br />
<math> {u_3} \geq 0 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Where <math> u_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3705Exponential transformation2021-11-27T21:43:30Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
(eq 1)<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t. <br />
<br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<br />
<math> x_1 \geq 0 </math><br />
<br />
<math> x_2 \geq 4 </math><br />
<br />
<math> x_3 \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
Using the exponential transformation described the problem becomes the following:<br />
<br />
min <math> Z = 5{e^{2{u_1}}}{e^{8{u_2}}} + 2{e^{u_1}}{e^{2u_2}} + {\frac{e^{{u_2}{3}}}{e^u_3}} + 5{y_1} + 2 {y_2^2} </math><br />
<br />
s.t<br />
<br />
<math> {e^{u_1}}{e^{4{e_3}}} \leq 7{e^{0.2{u_2}}} </math><br />
<br />
<math> 2{e^{3{u_1}}} - y_1^2 \leq 1 </math><br />
<br />
<math> e^{u_1} \geq 0 </math><br />
<br />
<math> e^{u_2} \geq 4 </math><br />
<br />
<math> e^{u_3} \geq 1 </math><br />
<br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
With additional logarithmic simplification:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
s.t. <br />
<math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<math> x_1 \geq 0 </math><br />
<math> x_2 \geq 4 </math><br />
<math> x_3 \geq 1 </math><br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
<br />
Where <math> x_1 </math> is unbounded due to logarithmic of 0 being indefinite.<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3700Exponential transformation2021-11-27T21:33:03Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
(eq 1)<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation 1. This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not transformed. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = 5{x_1^2}{x_2^8} + 2{x_1}{x_3^2} + {\frac{x_2^3}{x_3}} + 5{y_1} + 2 {y_2^2} </math><br />
s.t. <math> {x_1}{x_3^4} \leq 7{x_2^{0.2}} </math><br />
<math> 2{x_1^3} - y_1^2 \leq 1 </math><br />
<math> x_1 \geq 0 </math><br />
<math> x_2 \geq 4 </math><br />
<math> x_3 \geq 1 </math><br />
<math> y_1 = 0,1 </math> <math> y_2 = 0,1 </math> <br />
<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3697Exponential transformation2021-11-27T21:15:58Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP by allowing the use of a global solve, in some cases linearization can be achieved for certain constraints through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications in Computational Optimization ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. Using the exponential substitution detailed above all continuous variables in the function are transformed while binary variables are not <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = {5{x_1^2}{x_2^8} + 2{x1}{x_3^2} + {\frac{x_2^3}{x_3}] + 5{y_1} + 2 {y_2^2} <\math><br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:<br />
* As seen in eq(34) and (35) of the work by Björk and Westerlund they employ an exponential transformation to convexify their optimization problem to employ a global optimization approach. <ref>{{cite journal |title=Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption |journal=Computers & Chemical Engineering |year=2002 |last=Björk |first=Kaj-Mikael |last2=Westerlund |first2=Tapio |volume=26 |issue=11 |pages=1581-1593 |issn=ISSN 0098-1354 |doi=10.1016/S0098-1354(02)00129-1 |url=https://www.sciencedirect.com/science/article/pii/S0098135402001291?casa_token=G7OVOrBKagoAAAAA:1hUCHkGascVlawR3OfBpolNXlFqPSBUhWL6MkVAhn-ofKVfF-CbhVK6ZfSCKQ7i6mRQ9MTaqf9Q |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /><br />
<br />
{{reflist}}</div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3694Exponential transformation2021-11-27T20:59:24Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
min <math> Z = {5{x_1^2}{x_2^8} + 2{x1}{x_3^2} + {\frac{x_2^3}{x_3}] + 5{y_1} + 2 {y_2^2} <\math><br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in practical solutions. <br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric MINLP and obtain a global solution to the problem. Exponential transformation does not alter the bounds of the problem and allows for a convex objective function and constraints given that the conditions described under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3693Exponential transformation2021-11-27T20:50:05Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> Also presented by Li and Biswal the bounds of the problem are not altered through exponential transformation. <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
The following MINLP problem can take a Covexification approach using exponential transformation:<br />
<br />
<br />
<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<math> <br />
\begin{bmatrix} <br />
X & X \\<br />
X & Y<br />
\end{bmatrix}<br />
</math><br />
<br />
<br />
== Current Applications ==<br />
<br />
Currently various applications of exponential transformation can be seen in published journal articles and industry practices, due to the closeness with logarithmic transformation usually a combination of the approaches are used in <br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3686Exponential transformation2021-11-27T20:28:12Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref> and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
<br />
<br />
The transformed objective function can be show to be convex through the positive-definite test of the Hessian, for the example above the Hessian is as follows <ref> Chiang, Mung. (2005). Geometric Programming for Communication Systems. 10.1561/9781933019574; https://www.princeton.edu/~chiangm/gp.pdf </ref>:<br />
<br />
Proof of convexity of with positive definite test of Hessian <br />
<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3684Exponential transformation2021-11-27T20:21:13Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref>and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3683Exponential transformation2021-11-27T20:20:42Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref>and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math>{{e^{u_1}}^3}*{{e^{u_2}}^{-4}} + {{e^{u_1}}^2} + {{e^{u_2}}^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Simplifying by exponent properties<br />
<br />
<math> {e^{3{u_1} - 4{u_2}}} + {e^{{2}{u_1}}} + {{e^{{\frac{2}{3}}*{u_2}}}} \leq 4 </math><br />
<br />
Further linearization with natural logarithm<br />
<br />
<math>({{3}{u_1}-{4}{u_2}}) + {2}{u_1} + {\frac{2}{3}}{u_2} \leq \ln 4 </math><br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3679Exponential transformation2021-11-27T20:09:38Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref>and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math> {e^{{u_1}*{3}}}*{e^{{u_2}*{-4}}} + {e^{{u_2}*{2}}} + {e^{{u_2}*{\frac{2}{3}}} \leq 4 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
Linearization example<br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3677Exponential transformation2021-11-27T20:07:24Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref>and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math> {e^{u_1*3}}*{e^{{u_2}*{-4}}} + {e^{u_2*2} + {e^{{u_2}*{\frac{2}{3}}} \leq 4 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
Linearization <br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in "Global optimization of signomial geometric programming using linear relaxation" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref><br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3674Exponential transformation2021-11-27T19:59:33Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref>and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal (Positive and Polynomial) noncovex function of the form <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <ref> Grossmann, I.E. Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques. Optimization and Engineering 3, 227–252 (2002). https://doi.org/10.1023/A:1021039126272 </ref><br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
== Proof == <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in """Global optimization of signomial geometric programming using linear relaxation""" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref> Boyd, S., Kim, SJ., Vandenberghe, L. et al. A tutorial on geometric programming. Optim Eng 8, 67 (2007). https://doi.org/10.1007/s11081-007-9001-7 </ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math> {e^{u_1*3}}*{e^{{u_2}*{-4}}} + {e^{u_2*2} + {e^{{u_2}*{\frac{2}{3}}}} \leq 4 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
Linearization <br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3672Exponential transformation2021-11-27T19:51:37Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. They are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. This transformation creates a convex function without changing the decision space of the problem <ref> Li, D., Biswal, M.P. Exponential Transformation in Convexifying a Noninferior Frontier and Exponential Generating Method. Journal of Optimization Theory and Applications 99, 183–199 (1998). https://doi.org/10.1023/A:1021708412776 </ref>and reducing the time to solve an NLP/MINLP, in some cases linearization can be achieved through exponential transformation as seen in the example below.<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied [reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf]<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
== Proof == <br />
<br />
<br />
Additionally as presented in Theorem 1 and accompanying proof in """Global optimization of signomial geometric programming using linear relaxation""" by P. Shen, K. Zhang, given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math> {e^{u_1*3}}*{e^{{u_2}*{-4}}} + {e^{u_2*2} + {e^{{u_2}*{\frac{2}{3}}}} \leq 4 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
Linearization <br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application in MINLP ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3671Exponential transformation2021-11-27T19:38:25Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} \leq 4 </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<br />
<math> {e^{u_1*3}}*{e^{{u_2}*{-4}}} + {e^{u_2*2} + {e^{{u_2}*{\frac{2}{3}}}} \leq 4 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3670Exponential transformation2021-11-27T19:36:12Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents<br />
<br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3669Exponential transformation2021-11-27T19:35:31Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents: <br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\frac{2}{3}}} </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3668Exponential transformation2021-11-27T19:35:12Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
<br />
Reformulating to exponents: <br />
<math> {x_1^3}*{x_2^{-4}} + {x_1^2} + {x_2^{\fract{2}{3}}} </math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3667Exponential transformation2021-11-27T19:32:57Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{{u_1}{a_1k}}{e}^{{u_2}{a_2k}}....{e}^{{u_n}{a_nk}}} </math><br />
<br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
Reformulating to exponents: <br />
<math> {x_1^3}*{x_2^-4} + {x_1^2} + {x_2^(\fract{2}{3}}}</math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3666Exponential transformation2021-11-27T19:32:02Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e}^{u_1}{a_1k}}{e}^{u_2}{a_2k}....{e}^{{u_n}{a_nk}} </math><br />
<br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
Reformulating to exponents: <br />
<math> {x_1^3}*{x_2^-4} + {x_1^2} + {x_2^(\fract{2}{3}}}</math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
== Current Applications ==<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and obtain a global solution to the problem. However, this approach can only be applied given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3664Exponential transformation2021-11-27T19:25:52Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are simple algebraic transformation of monomial functions through a variable substitution with an exponential variable. <br />
<br />
They used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e_1}^{u}{a_1k}}{e_2}^{a_2k}....{e_n}^{a_nk} </math><br />
<br />
<br />
<br />
== Proof == <br />
<br />
As presented in Theorem 1 and accompanying proof in Global optimization of signomial geometric programming using linear relaxation given that a function is being minimized it shows that after transformation all points on the transformed function are feasible in the original function and all objective values in the transformed function are the same or less than the original function. <ref>{{cite journal |title=Global optimization of signomial geometricprogramming using linear relaxation |journal=Elsevier: Applied Mathematics and Computation |year=2004 |last=Shen |first=Peiping |last2=Zhang |first2=Kecun |volume=150 |issue=1 |pages=99-114 |issn=0096-3003 |doi=10.1016/S0096-3003(03)00200-5 |accessdate=2021-11-27 }}</ref><br />
<br />
<br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
Reformulating to exponents: <br />
<math> {x_1^3}*{x_2^-4} + {x_1^2} + {x_2^(\fract{2}{3}}}</math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
Example:<br />
<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a useful method to convexify Geometric NLP/MINLP and be able to solution the problem the solution approach but can only be done given certain parameters are met. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3662Exponential transformation2021-11-27T18:57:39Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
The transformed function is presented as:<br />
<br />
<math> f(u) = \sum_{k=1}^N c_k{{e_1}^{u}{a_1k}}{e_2}^{a_2k}....{e_n}^{a_nk} </math><br />
<br />
<br />
<br />
== Proof == <br />
<br />
<br />
== Numerical Example ==<br />
<math> {\frac{x_1^3}{x_2^4}} + {x_1^2} + {\sqrt[3]{x_2^2}} \leq 4</math><br />
Reformulating to exponents: <br />
<math> {x_1^3}*{x_2^-4} + {x_1^2} + {x_2^(\fract{2}{3}}}</math><br />
<br />
Substituting <math> x_1 = e^{u_1}, x_2 = e^{u_2} </math><br />
<math> {{e^u_1}}^3 </math><br />
<br />
<math> </math><br />
<math> </math><br />
<math> </math><br />
<br />
<br />
<br />
== Applications ==<br />
Exponential transformation can be used for convexification of any Geometric NLP or MINLP that meet the criteria of equation (1). This is done by turning the problem into a nonlinear convex optimization problem through exponential transformation. <br />
<br />
<br />
== Example of Convexification application ==<br />
<br />
<br />
Proof of convexity with positive definite test of Hessian <br />
<br />
<br />
https://www.princeton.edu/~chiangm/gp.pdf <br />
<br />
pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf<br />
<br />
Example:<br />
<br />
<br />
<br />
Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf<br />
<br />
<br />
Quadratic Geometric Programming<br />
Ecconomics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf <br />
<br />
== Conclusion ==<br />
Exponential transformation is a powerful method to convexify Geometric NLP/MINLP to simplify the solution approach. <br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3655Exponential transformation2021-11-27T18:04:21Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49, Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021<br />
<br />
<br />
== Introduction ==<br />
<br />
Exponential transformations are used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems. <br />
<br />
<br />
Geometric Programming <br />
<br />
<br />
== Theory & Methodology ==<br />
<br />
Exponential transformation begins with a posynominal noncovex function of the form <ref>{{cite news |last=Boyd |first=Stephen |last2=Kim |first2=Seung-Jean |last3=Vandenberghe |first3=Lieven |last4=Hassibi |first4=Arash |format=PDF |title=A tutorial on geometric programming |work=Springer Science+Business Media, LLC 2007 |publisher=Springer Science+Business Media, LLC 2007 |date=2007-04-10 |deadurl=no |archiveurl=https://web.stanford.edu/~boyd/papers/pdf/gp_tutorial.pdf }}</ref> :<br />
<br />
<math> f(x) = \sum_{k=1}^N c_k{x_1}^{a_1k}{x_2}^{a_2k}....{x_n}^{a_nk} </math><br />
<br />
where <math> c_k \geq 0 </math> and <math> x_n \geq0 </math><br />
<br />
A transformation of <math> x_n = e^u_i </math> is applied <br />
<br />
<br />
Transformed into convex MINLP <br />
<br />
<br />
<math><br />
\ln c<br />
</math><br />
<br />
== Proof == <br />
<br />
<br />
== Numerical Example ==<br />
<br />
== Applications ==<br />
<br />
<br />
<br />
== Conclusion ==<br />
Exponential transformation is a powerful method to linearize <br />
<br />
<br />
== References ==<br />
<references /></div>Dasogilhttps://optimization.cbe.cornell.edu/index.php?title=Exponential_transformation&diff=3152Exponential transformation2021-11-22T00:59:02Z<p>Dasogil: </p>
<hr />
<div>Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49, Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021</div>Dasogil