8.8 Solution of a Constrained Geometric Programming Problem 509
andakij (k= 1 , 2 ,... , m; i= 1 , 2 ,... , n;j= 1 , 2 ,... , Nk) re any real numbers,a
mindicates the total number of constraints,N 0 represents the number of terms in
the objective function, andNk denotes the number of terms in thekth constraint.
The design variables x 1 , x 2 ,... , xn are assumed to take only positive values in
Eqs. (8.52) and (8.53). The solution of the constrained minimization problem stated
above is considered in the next section.
8.8 Solution of a Constrained Geometric Programming Problem
For simplicity of notation, let us denote the objective function as
x 0 =g 0 ( X)=f(X)=
∑N^0
i= 1
c 0 j
∏n
j= 1
x
a 0 ji
i (8.54)
The constraints given in Eq. (8.53) can be rewritten as
fk=σk[1−gk(X)]≥ 0 , k= 1 , 2 ,... , m (8.55)
whereσk, the signum function, is introduced for thekth constraint so that it takes on
the value+1 or−1, depending on whethergk( X)is ≤1 or ≥1, respectively. The
problem is to minimize the objective function, Eq. (8.54), subject to the inequality
constraints given by Eq. (8.55). This problem is called theprimal problemand can be
replaced by an equivalent problem (known as thedual problem) with linear constraints,
which is often easier to solve. The dual problem involves the maximization of the dual
function,v(λ), given by
v(λ)=
∏m
k= 0
∏Nk
j= 1
(
ckj
λkj
∑Nk
l= 1
λkl
)σkλkj
(8.56)
subject to the normality and orthogonality conditions
∑N^0
j= 1
λ 0 j= 1 (8.57)
∑m
k= 0
∑Nk
j= 1
σkakijλkj= 0 , i= 1 , 2 ,... , n (8.58)
If the problem has zero degree of difficulty, the normality and orthogonality conditions
[Eqs. (8.57) and (8.58)] yield a unique solution forλ∗from which the stationary value
of the original objective function can be obtained as
f∗=x 0 ∗= v(λ∗)=
∏m
k= 0
∏Nk
j= 1
(
ckj
λ∗kj
∑Nk
l= 1
λ∗kl
)σkλ∗kj
(8.59)
