Engineering Optimization: Theory and Practice, Fourth Edition

8 Geometric Programming

8.1 Introduction

Geometric programming is a relatively new method of solving a class of nonlinear

programming problems. It was developed by Duffin, Peterson, and Zener [8.1]. It is

used to minimize functions that are in the form of posynomials subject to constraints of

the same type. It differs from other optimization techniques in the emphasis it places on

the relative magnitudes of the terms of the objective function rather than the variables.

Instead of finding optimal values of the design variables first, geometric programming

first finds the optimal value of the objective function. This feature is especially advan-

tageous in situations where the optimal value of the objective function may be all that

is of interest. In such cases, calculation of the optimum design vectors can be omitted.

Another advantage of geometric programming is that it often reduces a complicated

optimization problem to one involving a set of simultaneous linear algebraic equations.

The major disadvantage of the method is that it requires the objective function and

the constraints in the form of posynomials. We will first see the general form of a

posynomial.

8.2 Posynomial

In an engineering design situation, frequently the objective function (e.g., the total cost)

f(X) is given by the sum of several component costsUi( asX)

f (X)=U 1 +U 2 + · · · +UN (8.1)

Inmany cases, the component costsUican be expressed as power functions of the

type

Ui=ci xa 11 i x 2 a^2 i · · ·xnani (8.2)

where the coefficientsci are positive constants, the exponentsaijare real constants

(positive, zero, or negative), and the design parametersx 1 , x 2 ,... , xnare taken to be

positive variables. Functions likef, because of the positive coefficients and variables

and real exponents, are calledposynomials. For example,

f (x 1 , x 2 , x 3 )= 6 + 3 x 1 − 8 x 2 + 7 x 3 + 2 x 1 x 2

− 3 x 1 x 3 +^43 x 2 x 3 +^87 x 12 − 9 x^22 +x 32

492 Engineering Optimization: Theory and Practice, Fourth Edition Singiresu S. Rao
Copyright © 2009 by John Wiley & Sons, Inc.