Exponential transformation

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

Introduction

Exponential transformations are 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:

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

Numerical Example

${\frac {x_{1}^{3}}{x_{2}^{4}}}+{x_{1}^{2}}+{\sqrt[{3}]{x_{2}^{2}}}\leq 4$ Reformulating to exponents: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x_1^3}*{x_2^-4} + {x_1^2} + {x_2^(\fract{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

Example:

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 powerful method to convexify Geometric NLP/MINLP to simplify the solution approach.