8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 505Primal and Dual Problems. We saw that geometric programming treats the prob-
lem of minimizing posynomials and maximizing product functions. The minimization
problems are calledprimal programsand the maximization problems are calleddual
programs. Table 8.1 gives the primal and dual programs corresponding to an uncon-
strained minimization problem.
Computational Procedure. To solve a given unconstrained minimization problem,
we construct the dual functionv()and maximize eitherv()or lnv(), whichever
is convenient, subject to the constraints given by Eqs. (8.48) and (8.49). If the degree
of difficulty of the problem is zero, there will be a unique solution for the∗j’s.
For problems with degree of difficulty greater than zero, there will be more vari-
ablesj (j = 1 , 2 ,... , N )than the number of equations (n+1). Sometimes it will
be possible for us to express any (n+1) number ofj’s in terms of the remaining
(N−n− 1 ) number ofj’s. In such cases, our problem will be to maximizev()or
lnv()with respect to the (N−n−1) independentj’s. This procedure is illustrated
withthe help of the following one-degree-of-difficulty example.
Example 8.2 In a certain reservoir pump installation, the first cost of the pipe is given
by (100D+ 50 D^2 ) where, Dis the diameter of the pipe in centimeters. The cost of
the reservoir decreases with an increase in the quantity of fluid handled and is given
by 20/Q, whereQis the rate at which the fluid is handled (cubic meters per second).
Table 8.1 Primal and Dual Programs Corresponding to an Unconstrained
Minimization Problem
Primal program Dual programFindX=
x 1
x 2
..
.
xn
Find=
1
2
..
.
N
so that so thatf (X)=∑N
j= 1cjx
a 1 j
1 xa 2 j
2 ·· ·xanj
n v()=∏N
j= 1(
cj
j)j→minimum or
x 1 > 0 , x 2 > 0 ,... , xn> 0lnv()=ln[
∏N
j= 1(
cj
j)j]
→ maximum(8.47)subject to the constraints
∑N
j= 1j= 1 (8.48)
∑N
j= 1aijj= 0 , i= 1 , 2 ,... , n (8.49)