432 Nonlinear Programming III: Constrained Optimization Techniques
Figure 7.10 Penalty function methods:(a)exterior method;(b)interior method.It can be seen from Fig. 7.10athat the unconstrained minima ofφ(X,rk) onvergec
to the optimum pointX∗as the parameterrkis increased sequentially. On the other
hand, the interior method shown in Fig. 7.10bgives convergence as the parameterrk
is decreased sequentially.
There are several reasons for the appeal of the penalty function formulations. One
main reason, which can be observed from Fig. 7.10, is that the sequential nature of
the method allows a gradual or sequential approach to criticality of the constraints. In
addition, the sequential process permits a graded approximation to be used in analysis
of the system. This means that if the evaluation off andgj [and henceφ(X,rk)]
for any specified design vectorXis computationally very difficult, we can use coarse
approximations during the early stages of optimization (when the unconstrained minima
ofφkare far away from the optimum) and finer or more detailed analysis approximation
during the final stages of optimization. Another reason is that the algorithms for the
unconstrained minimization of rather arbitrary functions are well studied and generally
are quite reliable. The algorithms of the interior and the exterior penalty function
methods are given in Sections 7.13 and 7.15.7.13 Interior Penalty Function Method
As indicated in Section 7.12, in the interior penalty function methods, a new function
(φfunction) is constructed by augmenting a penalty term to the objective function. The
penalty term is chosen such that its value will be small at points away from the con-
straint boundaries and will tend to infinity as the constraint boundaries are approached.
Hence the value of theφfunction also “blows up” as the constraint boundaries are