Engineering Optimization: Theory and Practice, Fourth Edition

522 Geometric Programming

Solution Procedure.

1.Approximate each of the posynomialsQ(X)†by a posynomial term. Then all

the constraints in Eq. (8.71) can be expressed as a posynomial to be less than

or equal to 1. This follows because a posynomial divided by a posynomial

term is again a posynomial. Thus with this approximation, the problem reduces

to an ordinary geometric programming problem. To approximateQ(X) by a

single-term posynomial, we choose any

̃

X> 0 and let

Uj=qj(X) (8.75)

j=

qj(

̃

X)

Q(

̃

X)

(8.76)

whereqjdenotes thejth term of the posynomialQ(X). Thus we obtain, by

using the arithmetic–geometric inequality, Eq. (8.22),

Q(X)=

∑

j

qj(X)≥

∏

j

[

qj(X)

qj(

̃

X)

Q(

̃

X)

]qj (

̃

X /Q()

̃

X)

(8.77)

By using Eq. (8.74), the inequality (8.77) can be restated as

Q(X)≥

̃

Q(X,

̃

X)≡Q(

̃

X)

∏

i

(

xi

̃

xi

)∑j[bijqj(

̃

X /Q()

̃

X)]

(8.78)

where the equality sign holds true ifxi=

̃

xi. We can takeQ(X,

̃

X)as an

approximation forQ(X) at

̃

X.

2.At any feasible pointX(^1 ), replaceQk( in Eq. (8.71) by their approximationsX)

̃

Qk(X,X(^1 )) and solve the resulting ordinary geometric programming problem,

to obtain the next pointX(^2 ).

3 .By continuing in this way, we generate a sequence{X(α)} where, X(α+^1 )is an

optimal solution for theαth ordinary geometric programming problem (OGPα):

Minimizex 0

subject to

Pk(X)

̃

Qk(X,X(α))

≤ 1 , k= 1 , 2 ,... , m

X=















x 0

x 1

x 2

..

.

xn















> 0 (8.79)

It has been proved [8.4] that under certain mild restrictions, the sequence of

points{X(α)} converges to a local minimum of the complementary geometric

programming problem.

†The subscriptkis removed forQ(X) for simplicity.