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 used for convexification of geometric programming constraints (posynominal) nonconvex optimization problems.

Geometric Programming

Theory & Methodology

Exponential transformation begins with a posynominal noncovex function of the form  :

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

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

A transformation of $x_{n}=e_{i}^{u}$ is applied reference https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf

The transformed function is presented as:

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

Proof

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.

Numerical Example

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

Reformulating to exponents: ${x_{1}^{3}}*{x_{2}^{-4}}+{x_{1}^{2}}+{x_{2}^{\frac {2}{3}}}$

Substituting $x_{1}=e^{u_{1}},x_{2}=e^{u_{2}}$ ${e_{1}^{u}}^{3}$

${\displaystyle }$ ${\displaystyle }$ ${\displaystyle }$

Applications

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.

Example of Convexification application

Proof of convexity with positive definite test of Hessian

https://www.princeton.edu/~chiangm/gp.pdf

pROOF THAT CHANGING IT DOESNT CHANGE THE BOUNDS OF THE PROBLEM https://link.springer.com/content/pdf/10.1023/A:1021708412776.pdf

Current Applications

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 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.