Exponential transformation

Author: Daphne Duvivier, Daniela Gil, Jacqueline Jackson, Sinclaire Mills, Vanessa Nobre (SYSEN 5800, Fall 2021)

Introduction

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. 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.

Theory & Methodology

Exponential transformation can be applied to geometric programs. A geometric program is a mathematical optimization problem. Geometric Programs take the form of:

${\begin{aligned}\min &\quad f_{0}(x)\\s.t.&\quad f_{i}(x)\leq 1i=1,....,m\\&\quad g_{i}(x)=1i=1,....,p\end{aligned}}$

convexification of a non-inferior frontier - based on having a differentiable objective function https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf

this would apply to any polynomial. A posynomial is defined as define polynomials:

Exponential transformation begins with a posynomial (Positive and Polynomial) noncovex function of the form  :

$f(x)=\sum _{k=1}^{N}c_{k}{x_{1}}^{a_{1k}}{x_{2}}^{a_{2k}}....{x_{n}}^{a_{nk}}$

where $c_{k}\geq 0$ and $x_{n}\geq 0$

A transformation of $x_{n}=e_{i}^{u}$ is applied

The transformed function is presented as:

$f(u)=\sum _{k=1}^{N}c_{k}{{e}^{{u_{1}}{a_{1k}}}{e}^{{u_{2}}{a_{2k}}}....{e}^{{u_{n}}{a_{nk}}}}$

Exponential Transformation in Computational Optimization

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.

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. Also presented by Li and Biswal, the bounds of the problem are not altered through exponential transformation.

Numerical Example

${\frac {x_{1}^{3}}{x_{2}^{4}}}+{x_{1}^{2}}+{\sqrt[{3}]{x_{2}^{2}}}\leq 4$

Reformulating to exponents

${x_{1}^{3}}*{x_{2}^{-4}}+{x_{1}^{2}}+{x_{2}^{\frac {2}{3}}}\leq 4$

Substituting $x_{1}=e^{u_{1}},x_{2}=e^{u_{2}}$

${e^{{3}{u_{1}}}}*{e^{{-4}{u_{2}}}}+{e^{{2}{u_{1}}}}+{{e^{u_{2}}}^{\frac {2}{3}}}\leq 4$

Simplifying by exponent properties

${e^{3{u_{1}}-4{u_{2}}}}+{e^{{2}{u_{1}}}}+{e^{{\frac {2}{3}}{u_{2}}}}\leq 4$

Example of Convexification Application in MINLP

The following MINLP problem can take a convexification approach using exponential transformation:

${\begin{aligned}\min &\quad Z=5{x_{1}^{2}}{x_{2}^{8}}+2{x_{1}}+{\frac {x_{2}^{3}}{+}}5{y_{1}}+2{y_{2}^{2}}\\s.t.&\quad {x_{1}}\leq 7{x_{2}^{0.2}}\\&\quad 2{x_{1}^{3}}-y_{1}^{2}\leq 1\\&\quad x_{1}\geq 0\\&\quad x_{2}\leq 4\\&\quad y_{1}=0,1\quad y_{2}=0,1\end{aligned}}$

Using the exponential transformation to continuous variables $x_{1},x_{2},x_{3}$ by substituting $x_{1}=e^{u_{1}}$ and $x_{2}=e^{u_{2}}$ described the problem becomes the following:

${\begin{aligned}\min &\quad Z=5{e^{2{u_{1}}}}{e^{8{u_{2}}}}+2{e^{u_{1}}}{e^{2u_{2}}}+{\frac {e^{{u_{2}}{3}}}{+}}5{y_{1}}+2{y_{2}^{2}}\\s.t.&\quad {e^{u_{1}}}\leq 7{e^{0.2{u_{2}}}}\\&\quad 2{e^{3{u_{1}}}}-y_{1}^{2}\leq 1\\&\quad e^{u_{1}}\geq 0\\&\quad e^{u_{2}}\leq 4\\&\quad y_{1}=0,1\quad y_{2}=0,1\end{aligned}}$

With additional logarithmic simplification through properties of natural logarithm:

${\begin{aligned}\min &\quad Z=5{e^{2{u_{1}}}}{e^{8{u_{2}}}}+2{e^{u_{1}}}{e^{2u_{2}}}+{\frac {e^{{u_{2}}{3}}}{+}}5{y_{1}}+2{y_{2}^{2}}\\s.t.&\quad u_{1}\leq \ln 7+0.2{u_{2}}\\&\quad 2{e^{3{u_{1}}}}-y_{1}^{2}\leq 1\\&\quad {u_{2}}\leq \ln 4\\&\quad y_{1}=0,1\quad y_{2}=0,1\end{aligned}}$

Where $u_{1}$ is unbounded due to logarithmic of 0 being indefinite.

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 :

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 . This tests the Hessian defined as:

$H(x)=H=\nabla ^{2}f(x)$

to test that

$Q(x)\geq 0$

where

$Q(x)=x^{T}Hx$

for all $x\neq 0$

for functions the Hessian is defined as:

${\begin{bmatrix}X&X\\X&Y\end{bmatrix}}$

In the example above, the Hessian is defined as:

${\begin{bmatrix}X&X\\X&Y\end{bmatrix}}$

Therefore, H(x) is positive-definite and strictly convex.

Applications

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.

Global optimization of heat exchanger network synthesis problems with and without the isothermal mixing assumption:

• 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.

Electrical Engineering Application: http://home.eng.iastate.edu/~cnchu/pubs/j08.pdf

Quadratic Geometric Programming Economics: https://link.springer.com/content/pdf/10.1007/BF02591746.pdf

Conclusion

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 are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.