7.15 Exterior Penalty Function Method 443
SUMT method is a global one. In such cases one has to satisfy with a local minimum
only. However, one can always reapply the SUMT method from different feasible
starting points and try to find a better local minimum point if the problem has several
local minima. Of course, this procedure requires more computational effort.
7.15 Exterior Penalty Function Method
In the exterior penalty function method, theφfunction is generally taken as
φ(X, rk) =f(X)+rk
∑m
j= 1
〈gj(X)〉q (7.199)
whererkis a positive penalty parameter, the exponentqis a nonnegative constant, and
the bracket function〈gj( X)〉is defined as
〈gj( X)〉=max〈gj( X), 0 〉
=
gj( X) ifgj(X)> 0
(constraint is violated)
0 ifgj(X)≤ 0
(constraint is satisfied)
(7.200)
It can be seen from Eq. (7.199) that the effect of the second term on the right side is to
increaseφ(X, rk) n proportion to thei qth power of the amount by which the constraints
are violated. Thus there will be a penalty for violating the constraints, and the amount of
penalty will increase at a faster rate than will the amount of violation of a constraint (for
q>1). This is the reason why the formulation is called the penalty function method.
Usually, the functionφ(X, rk) ossesses a minimum as a function ofp Xin the infeasible
region. The unconstrained minimaX∗kconverge to the optimal solution of the original
problem ask→ ∞andrk→ ∞ .Thus the unconstrained minima approach the feasible
domain gradually, and ask→ ∞, theX∗keventually lies in the feasible region. Let us
consider Eq. (7.199) for various values ofq.
1.q=0. Here theφfunction is given by
φ(X, rk) =f(X)+rk
∑m
j= 1
〈gj(X)〉^0
=
{
f(X)+mrk if allgj(X)> 0
f(X) if allgj(X)≤ 0
(7.201)
This function is discontinuous on the boundary of the acceptable region as
shown in Fig. 7.11 and hence it would be very difficult to minimize this function.
- 0 < q <1. Here theφfunction will be continuous, but the penalty for violating
a constraint may be too small. Also, the derivatives of the function are discon-
tinuous along the boundary. Thus it will be difficult to minimize theφfunction.
Typical contours of theφfunction are shown in Fig. 7.12.
