Engineering Optimization: Theory and Practice, Fourth Edition

502 Geometric Programming

In this section we prove thatf∗=v∗ and also thatf∗corresponds to the global

minimum off (X). For convenience of notation, let us denote the objective function

f (X)byx 0 and make the exponential transformation

ewi=xi or wi= nl xi, i= 0 , 1 , 2 ,... , n (8.30)

where the variableswi are unrestricted in sign. Define the new variablesj, also

termedweights, as

j=

Uj

x 0

=

cj

∏n

i= 1

x

aij

i

x 0

, j= 1 , 2 ,... , N (8.31)

which can be seen to be positive and satisfy the relation

∑N

j= 1

j= 1 (8.32)

By taking logarithms on both sides of Eq. (8.31), we obtain

lnj= nl cj+

∑n

i= 1

aijlnxi− nl x 0 (8.33)

or

ln

j

cj

=

∑n

i= 1

aijwi−w 0 , j= 1 , 2 ,... , N (8.34)

Thus the original problem of minimizingf (X)with no constraints can be replaced

by one of minimizingw 0 subject to the equality constraints given by Eqs. (8.32) and

(8.34). The objective functionx 0 is given by

x 0 =ew^0 =

∑N

j= 1

cj

∏n

i= 1

eaijwi

=

∑N

j= 1

cje

∑n

i= 1 aijwi (8.35)

Since the exponential function (eaijwi) is convex with respect towi, the objective

functionx 0 , which is a positive combination of exponential functions, is also convex

(see Problem 8.15). Hence there is only one stationary point forx 0 and it must be the

global minimum. The global minimum point ofw 0 can be obtained by constructing the

following Lagrangian function and finding its stationary point:

L(w,,λ)=w 0 +λ 0

(N

∑

i= 1

i− 1

)

+

∑N

j= 1

λj

(n

∑

i= 1

aijwi−w 0 − nl

j

cj

)

(8.36)