Engineering Optimization: Theory and Practice, Fourth Edition

518 Geometric Programming

To find the maximum ofv, we set the derivative ofvwith respect toλ 01 equal to

zero. To simplify the calculations, we setd (lnv)/dλ 01 = and find the value of 0 λ∗ 01.

Then the values ofλ∗ 02 , λ∗ 03 , λ∗ 11 , λ∗ 12 , andλ∗ 21 can be found from Eqs. (E 14 ) o (Et 18 ).

Once the dual variables (λ∗kj) re known, Eqs. (8.62) and (8.63) can be used to finda

the optimum values of the design variables as in Example 8.3.

8.10 Geometric Programming with Mixed Inequality Constraints

In this case the geometric programming problem contains at least one signum function

with a value ofσk= − 1 amongk= 1 , 2 ,... , m. (Note thatσ 0 = + 1 corresponds to

the objective function.) Here no general statement can be made about the convexity

or concavity of the constraint set. However, since the objective function is continuous

and is bounded below by zero, it must have a constrained minimum provided that there

exist points satisfying the constraints.

Example 8.5

Minimizef =x 1 x^22 x 3 −^1 + 2 x 1 −^1 x− 23 x 4 + 01 x 1 x 3

subject to

3 x 1 x 3 −^1 x 42 + 4 x− 31 x 4 −^1 ≥ 1

5 x 1 x 2 ≤ 1

SOLUTION In this problem,m= 2 , N 0 = 3 ,N 1 = 2 ,N 2 = 1 ,N= 6 , n=4, and the

degree of difficulty is 1. The signum functions areσ 0 = , 1 σ 1 = − 1 , andσ 2 =. The 1

dual objective function can be stated, using Eq. (8.56), as follows:

Maximizev(λ)=

∏^2

k= 0

∏Nk

j= 1

(

ckj

λkj

∑Nk

l= 1

λkl

)σkλkj

=

[

c 01

λ 01

(λ 01 +λ 02 +λ 03 )

]λ 01 [

c 02

λ 02

(λ 01 +λ 02 +λ 03 )

]λ 02

×

[

c 03

λ 03

(λ 01 +λ 02 +λ 03 )

]λ 03

×

[

c 11

λ 11

(λ 11 +λ 12 )

]−λ 11 [

c 12

λ 12

(λ 11 +λ 12 )

]−λ 12 (

c 21

λ 21

λ 21

)λ 21

=

(

1

λ 01

)λ 01 (

2

λ 02

)λ 02 (

10

λ 03

)λ 03 [

3 (λ 11 +λ 12 )

λ 11

]−λ 11

×

[

4 (λ 11 +λ 12 )

λ 12

]−λ 12

( 5 )λ^21 (E 1 )