Engineering Optimization: Theory and Practice, Fourth Edition

8.4 Solution Using Differential Calculus 493

is a second-degree polynomial in the variables,x 1 , x 2 , andx 3 (coefficients of the various

terms are real) while

g(x 1 , x 2 , x 3 )=x 1 x 2 x 3 +x^21 x 2 + 4 x 3 +

2

x 1 x 2

+ 5 x

− 1 / 2

3

is a posynomial. If the natural formulation of the optimization problem does not lead to

posynomial functions, geometric programming techniques can still be applied to solve

the problem by replacing the actual functions by a set of empirically fitted posynomials

over a wide range of the parametersxi.

8.3 Unconstrained Minimization Problem

Consider the unconstrained minimization problem:

FindX=











x 1

x 2

..

.

xn











that minimizes the objective function

f (X)=

∑N

j= 1

Uj(X)=

∑N

j= 1

(

cj

∏n

i= 1

x

aij

i

)

=

∑N

j= 1

(cjx

a 1 j

1 x

a 2 j

2 · · ·x

anj

n )^ (8.3)

wherecj> 0 ,xi> 0 , and theaijare real constants.

The solution of this problem can be obtained by various procedures. In the fol-

lowing sections, two approaches—one based on the differential calculus and the other

based on the concept of geometric inequality—are presented for the solution of the

problem stated in Eq. (8.3).

8.4 SOLUTION OF AN UNCONSTRAINED GEOMETRIC
PROGRAMMING PROGRAM USING
DIFFERENTIAL CALCULUS
According to the differential calculus methods presented in Chapter 2, the necessary
conditions for the minimum off are given by

∂f

∂xk

=

∑N

j= 1

∂Uj

∂xk

=

∑N

j= 1

(cj x

a 1 j

1 x

a 2 j

2 · · ·x

ak − 1 ,j

k− 1 akjx

akj− 1

k a

ak + 1 ,j

k+ 1 · · ·x

anj

n )=^0 ,

k= 1 , 2 ,... , n (8.4)