# Exponential transformation

Author: Daniel Garcia (ChBE 345)

Stewards: Dajun Yue and Prof. Fengqi You

Date presented: May 25, 2014

This article concerns the exponential transformation method for globally solving posynomial (or general geometric/signomial) optimization problems with nonconvex objective functions or constraints. A discussion of the method's development, use, and limitations will be presented.

## Background

Before discussing methods to solve posynomial optimization problems, a brief review of posynomials is of use. A posynomial, as defined by Duffin, Peterson, and Zener (1967) as a function of the form

$f(x_1,x_2,...,x_n)=\sum_{k=1}^K c_kx_1^{a_{1k}}...x_n^{a_{nk}}$

where the variables $x_i$ and the coefficients $c_k$ are positive, real numbers, and all of the exponents $a_{ik}$ are real numbers. For example,

$f(x_1,x_2)=14x_1^{3.2}x_2^{4}+x_1^2x_2^7$

is a posynomial in two variables, and

$f(x_1)=1500x_1^{3/5}$

is a posynomial with one variable.

It is not difficult to imagine a posynomial that is nonconvext, such as the examples above. Unfortunately, this can cause some problems when attempting to find a globally optimal solution of a posynomial program, as it is known that only convex problems guarantee a globally optimal solution.

Posynomial or geometric programming has been applied to solve problems in varied fields, such as signal circuit design, engineering design, project management, and inventory management, just to name a few. Clearly, the solution of such problems are important to the chemical engineer, and being able to globally solve such problems will equip the engineer with a powerful tool to solve a myriad of problems.

### Development

Many researchers attempted to solve such problems starting in the 1960's and 1970's. Methods used in the day aimed to find only locally optimal solutions, and employed methods such as successive approximation of posynomials (called "condensation"), so-called "psuedo-duality" methods which use a weaker form of duality, and adapted nonlinear programming methods. While locally optimal solutions are certainly better than no solution at all, the desire to find a globally optimal solution was strong enough to spur the development of other methods for posynomial programs in the 1990's. Such methods included global optimization algorithms based on exponential variable transformations of the original posynomial program, convex relaxation of the original problem, and branch-and-bound-type methods.

### Historical Use

Give some examples here.

## Limitations

Discuss limitations here

## Conclusions

Content under conclusions