Engineering Optimization: Theory and Practice, Fourth Edition

504 Geometric Programming

By substituting Eq. (8.45) into Eq. (8.36), we obtain

L(,w)= −

∑N

j= 1

jln

j

cj

+( 1 −w 0 )





∑N

j= 1

j− 1



+

∑n

i= 1

wi





∑N

j= 1

aijj





(8.46)

The function given in Eq. (8.46) can be considered as the Lagrangian function cor-

responding to a new optimization problem whose objective functionv( ̃)is given by

v( ̃)= −

∑N

j= 1

jln

j

cj

= nl





∏N

j= 1

(

cj

j

)j



 (8.47)

and the constraints by

∑N

j= 1

j− 1 = 0 (8.48)

∑N

j= 1

aijj= 0 , i= 1 , 2 ,... , n (8.49)

This problem will be the dual for the original problem. The quantities (1−w 0 ),

w 1 , w 2 ,... , wncan be regarded as the Lagrange multipliers for the constraints given

by Eqs. (8.48) and (8.49).

Now it is evident that the vectorwhich makes the Lagrangian of Eq. (8.46)

stationary will automatically give a stationary point for that of Eq. (8.36). It can be

proved that the function

jln

j

cj

, j= 1 , 2 ,... , N

is convex (see Problem 8.16) sincej is positive. Since the functionv ̃()is given

by the negative of a sum of convex functions, it will be a concave function. Hence

the functionv( ̃)will have a unique stationary point that will be its global maximum

point. Hence the minimum of the original primal function is same as the maximum of

the function given by Eq. (8.47) subject to the normality and orthogonality conditions

given by Eqs. (8.48) and (8.49) with the variablesjconstrained to be positive.

By substituting the optimal solution∗, the optimal value of the objective function

becomes

v ̃∗= ̃ v(∗ )=L(w∗,∗)=w∗ 0 = L(w∗,∗,λ∗)

=−

∑N

j= 1

∗jln

∗j

cj

(8.50)

By taking the exponentials and using the transformation relation (8.30), we get

f∗=

∏N

j= 1

(

cj

∗j

)∗j

(8.51)