Engineering Optimization: Theory and Practice, Fourth Edition

510 Geometric Programming

If the functionf(X) is known to possess a minimum, the stationary valuef∗given

by Eq. (8.59) will be the global minimum off since, in this case, there is a unique

solution forλ∗.

The degree of difficulty of the problem (D) is defined as

D=N−n− 1 (8.60)

whereNdenotes the total number of posynomial terms in the problem:

N=

∑m

k= 0

Nk (8.61)

If the problem has a positive degree of difficulty, the linear Eqs. (8.57) and (8.58)

can be used to express any (n+1) of theλkj’s in terms of the remainingDof the

λkj’s. By using these relations,vcan be expressed as a function of theDindependent

λkj’s. Now the stationary points ofvcan be found by using any of the unconstrained

optimization techniques.

If calculus techniques are used, the first derivatives of the functionvwith respect

to the independent dual variables are set equal to zero. This results in as many simul-

taneous nonlinear equations as there are degrees of difficulty (i.e.,N−n−1). The

solution of these simultaneous nonlinear equations yields the best values of the dual

variables,λ∗. Hence this approach is occasionally impractical due to the computa-

tions required. However, if the set of nonlinear equations can be solved, geometric

programming provides an elegant approach.

Optimum Design Variables. For problems with a zero degree of difficulty, the solu-

tion ofλ∗is unique. Once the optimum values ofλkjare obtained, the maximum of the

dual functionv∗can be obtained from Eq. (8.59), which is also the minimum of the pri-

mal function,f∗. Once the optimum value of the objective functionf∗=x∗ 0 is known,

the next step is to determine the values of the design variablesxi∗(i = 1 , 2 ,... , n).

This can be achieved by solving simultaneously the following equations:

∗ 0 j=λ∗ 0 j≡

c 0 j

∏n

i= 1

(xi∗)a^0 ij

x 0 ∗

, j= 1 , 2 ,... , N 0 (8.62)

∗kj=

λ∗kj

∑Nk

l= 1

λ∗kl

=ckj

∏n

i= 1

(x∗i)akij, j= 1 , 2 ,... , Nk (8.63)

k= 1 , 2 ,... , m

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities

If the original problem has a zero degree of difficulty, the minimum of the primal

problem can be obtained by maximizing the corresponding dual function. Unfortunately,

this cannot be done in the general case where there are some greater than type of

inequality constraints. However, if the problem has all the constraints in the form of