7.13 Interior Penalty Function Method 4355.If all the constraints are not satisfied at the pointXM, set the new starting point
asX 1 =XM, and renumber the constraints such that the lastrconstraints will
be the unsatisfied ones (this value ofrwill be different from the previous value),
and go to step 2.
This procedure is repeated until all the constraints are satisfied and a pointX 1 =
XMis obtained for whichgj(X 1 ) < 0 ,j= 1 , 2 ,... , m.
If the constraints are consistent, it should be possible to obtain, by applying the
procedure, a pointX 1 that satisfies all the constraints. However, there may exist situa-
tions in which the solution of the problem formulated in step 3 gives the unconstrained
or constrained local minimum ofgk( X)that is positive. In such cases one has to start
afresh with a new pointX 1 from step 1 onward.
Initial Value of the Penalty Parameter (r 1 ). Since the unconstrained minimization
ofφ(X,rk) s to be carried out for a decreasing sequence ofi rk, it might appear that by
choosing a very small value ofr 1 , we can avoid an excessive number of minimizations
of the functionφ. But from a computational point of view, it will be easier to minimize
the unconstrained functionφ(X,rk) fi rkis large. This can be seen qualitatively from
Fig.7.10b. As the value ofrkbecomes smaller, the value of the functionφchanges
more rapidly in the vicinity of the minimumφk∗. Since it is easier to find the minimum of
afunction whose graph is smoother, the unconstrained minimization ofφwill be easier
ifrkis large. However, the minimum ofφk,X∗k, will be farther away from the desired
minimumX∗ifrk is large. Thus it requires an excessive number of unconstrained
minimizations ofφ(X,rk) (for several values ofrk) to reach the pointX∗ifr 1 is
selected to be very large. Thus a moderate value has to be choosen for the initial
penalty parameter (r 1 ) In practice, a value of. r 1 that gives the value ofφ(X 1 , r 1 )
approximatelyequal to 1.1 to 2.0 times the value off (X 1 ) as been found to be quiteh
satisfactory in achieving quick convergence of the process. Thus for any initial feasible
starting pointX 1 , the value ofr 1 can be taken as
r 1 ≃ 0. 1 to 1. 0f (X 1 )
−∑m
j= 11 g/j(X^1 )(7.164)
Subsequent Values of the Penalty Parameter. Once the initial value ofrkis chosen,
the subsequent values ofrk+ 1 have to be chosen such that
rk+ 1 < rk (7.165)For convenience, the values ofrkare chosen according to the relation
rk+ 1 = crk (7.166)wherec <1. The value ofccan be taken as 0.1, 0.2, or 0.5.
Convergence Criteria. Since the unconstrained minimization ofφ(X,rk) as to beh
carried out for a decreasing sequence of valuesrk, it is necessary to use proper con-
vergence criteria to identify the optimum point and to avoid an unnecessarily large
number of unconstrained minimizations. The process can be terminated whenever the
following conditions are satisfied.