Engineering Optimization: Theory and Practice, Fourth Edition

520 Geometric Programming

By using Eqs. (E 3 ) the dual objective function of Eq. (E, 1 ) an be expressed asc

v(λ 01 )=

(

1

λ 01

)λ 01 (

2

8 λ 01 − 4

) 8 λ 01 − 4 (

10

− 9 λ 01 + 5

) 5 − 9 λ 01

×

[

3 ( 10 λ 01 − 5 )

6 λ 01 − 3

]− 6 λ 01 + 3 [

4 ( 10 λ 01 − 5 )

4 λ 01 − 2

]− 4 λ 01 + 2

( 5 )^22 λ^01 −^21

=

(

1

λ 01

)λ 01 (

1

4 λ 01 − 2

) 8 λ 01 − 4 (

10

5 − 9 λ 01

) 5 − 9 λ 01

( 5 )^3 −^6 λ^01 ( 10 )^2 −^4 λ^01

×( 5 )^22 λ^01 −^21

=

(

1

λ 01

)λ 01 (

1

4 λ 01 − 2

) 8 λ 01 − 4 (

10

5 − 9 λ 01

) 5 − 9 λ 01

( 5 )^12 λ^01 −^7 ( 2 )^2 −^4 λ^01

To maximizev, setd(lnv)/dλ 01 = and find 0 λ∗ 01. Onceλ∗ 01 is known,λ∗kjcan be

obtained from Eqs. (E 3 ) andthe optimum design variables from Eqs. (8.62) and (8.63).

8.11 Complementary Geometric Programming

Avriel and Williams [8.4] extended the method of geometric programming to include

any rational function of posynomial terms and called the methodcomplementary geo-

metric programming.†The case in which some terms may be negative will then become

aspecial case of complementary geometric programming. While geometric program-

ming problems have the remarkable property that every constrained local minimum is

also a global minimum, no such claim can generally be made for complementary geo-

metric programming problems. However, in many practical situations, it is sufficient

to find a local minimum.

The algorithm for solving complementary geometric programming problems con-

sists of successively approximating rational functions of posynomial terms by posyn-

omials. Thus solving a complementary geometric programming problem by this algo-

rithm involves the solution of a sequence of ordinary geometric programming problems.

It has been proved that the algorithm produces a sequence whose limit is a local

minimum of the complementary geometric programming problem (except in some

pathological cases).

Let the complementary geometric programming problem be stated as follows:

MinimizeR 0 (X)

subjectto

Rk( X)≤ 1 , k= 1 , 2 ,... , m

where

Rk(X)=

Ak(X)−Bk(X)

Ck(X)−Dk(X)

, k= 0 , 1 , 2 ,... , m (8.66)

†The application of geometric programming to problems involving generalized polynomial functions was

presented by Passy and Wilde [8.2].