756 Practical Aspects of Optimization
14.8.2 Method
Let the optimization problem be stated as follows:FindX= {x 1 x 2 · · ·xn}Twhich minimizesf(X) (14.89)subject togj(X)≤ 0 , j= 1 , 2 ,... , m (14.90)hk(X)= 0 , k= 1 , 2 ,... , p (14.91)xi(l)≤xi≤xi(u), i= 1 , 2 ,... , n (14.92)wherex(l)i andx(u)i denote the lower and upper bounds onxi. Most systems permit the
partitioning of the vectorXinto two subvectorsYandZ:X=
{
Y
Z
}
(14.93)
where the subvectorYdenotes the coordination or interaction variables between the
subsystems and the subvectorZindicates the variables confined to subsystems. The
vectorZ, in turn, can be partitioned asZ=
Z 1
..
.
Zk
..
.
ZK
(14.94)
whereZkrepresents the variables associated with thekth subsystem only andKdenotes
the number of subsystems. The partitioning of variables, Eq. (14.94), permits us to
regroup the constraints as
g 1 (X)
g 2 (X)
..
.
gm(X)
=
g(^1 )(Y,Z 1 )
g(^2 )(Y,Z 2 )
..
.
g(K)(Y,ZK)
≤ 0 (14.95)
l 1 (X)
l 2 (X)
..
.
lp(X)
=
l(^1 )(Y,Z 1 )
l(^2 )(Y,Z 2 )
..
.
l(K)(Y,ZK)
= 0 (14.96)
where the variablesYmay appear in all the functions while the variablesZkappear
only in the constraint setsg(k)≤ 0 and h(k)=. The bounds on the variables, 0
Eq. (14.92), can be expressed as