Exponential transformation

Author: Daphne Duvivier (dld237), Daniela Gil (dsg254), Jacqueline Jackson (jkj49), Sinclaire Mills (sm2795), Vanessa Nobre (vmn28) Fall 2021

Introduction

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 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.

Theory & Methodology

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

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

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:

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

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

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

Simplifying by exponent properties

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

Further linearization with natural logarithm

$({{3}{u_{1}}-{4}{u_{2}}})+{2}{u_{1}}+{\frac {2}{3}}{u_{2}}\leq \ln 4$

Applications in Computational Optimization

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.

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.

Example of Convexification application in MINLP

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

min $Z=5{x_{1}^{2}}{x_{2}^{8}}+2{x_{1}}{x_{3}^{2}}+{\frac {x_{2}^{3}}{x_{3}}}+5{y_{1}}+2{y_{2}^{2}}$

s.t.

${x_{1}}{x_{3}^{4}}\leq 7{x_{2}^{0.2}}$

$2{x_{1}^{3}}-y_{1}^{2}\leq 1$

$x_{1}\geq 0$

$x_{2}\leq 4$

$x_{3}\geq 1$

$y_{1}=0,1$ $y_{2}=0,1$

Using the exponential transformation described the problem becomes the following:

min $Z=5{e^{2{u_{1}}}}{e^{8{u_{2}}}}+2{e^{u_{1}}}{e^{2u_{2}}}+{\frac {e^{{u_{2}}{3}}}{e_{3}^{u}}}+5{y_{1}}+2{y_{2}^{2}}$

s.t

${e^{u_{1}}}{e^{4{e_{3}}}}\leq 7{e^{0.2{u_{2}}}}$

$2{e^{3{u_{1}}}}-y_{1}^{2}\leq 1$

$e^{u_{1}}\geq 0$

$e^{u_{2}}\leq 4$

$e^{u_{3}}\geq 1$

$y_{1}=0,1$ $y_{2}=0,1$

With additional logarithmic simplification:

min $Z=5{e^{2{u_{1}}}}{e^{8{u_{2}}}}+2{e^{u_{1}}}{e^{2u_{2}}}+{\frac {e^{{u_{2}}{3}}}{e_{3}^{u}}}+5{y_{1}}+2{y_{2}^{2}}$

s.t

$u_{1}+4{u_{3}}\leq \ln 7+0.2{u_{2}}$

$2{e^{3{u_{1}}}}-y_{1}^{2}\leq 1$

${u_{2}}\leq \ln 4$

${u_{3}}\geq 0$

$y_{1}=0,1$ $y_{2}=0,1$

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

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 :

Proof of convexity of with positive definite test of Hessian

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

Current 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 Ecconomics: 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 under the Theory and methodology section are satisfied. Geometric Programming transformation can be further explored through logarithmic transformation to address convexification.