Engineering Optimization: Theory and Practice, Fourth Edition

7.13 Interior Penalty Function Method 433

approached. This behavior can also be seen from Fig. 7.10b. Thus once the uncon-
strained minimization ofφ(X,rk) s started from any feasible pointi X 1 , the subsequent
points generated will always lie within the feasible domain since the constraint bound-
aries act as barriers during the minimization process. This is why the interior penalty
function methods are also known asbarrier methods. Theφfunction defined originally
by Carroll [7.14] is

φ(X, rk) =f(X)−rk

∑m

j= 1

1

gj(X)

(7.160)

It can be seen that the value of the functionφwill always be greater thanfsincegj(X)
is negative for all feasible pointsX. If any constraintgj( X)is satisfied critically (with
equality sign), the value ofφtends to infinity. It is to be noted that the penalty term in
Eq. (7.160) is not defined ifXis infeasible. This introduces serious shortcoming while
using the Eq. (7.160). Since this equation does not allow any constraint to be violated,
it requires a feasible starting point for the search toward the optimum point. However,
in many engineering problems, it may not be very difficult to find a point satisfying
all the constraints,gj( X)≤0,at the expense of large values of the objective function,
f (X). If there is any difficulty in finding a feasible starting point, the method described
in the latter part of this section can be used to find a feasible point. Since the initial
point as well as each of the subsequent points generated in this method lies inside the
acceptable region of the design space, the method is classified as aninterior penalty
function formulation. Since the constraint boundaries act as barriers, the method is also
known as a barrier method. The iteration procedure of this method can be summarized
as follows.

Iterative Process
1.Start with an initial feasible pointX 1 satisfying all the constraints with strict
inequality sign, that is,gj(X 1 ) < 0 forj= 1 , 2 ,... , m, and an initial value of
r 1 > 0. Setk=1.
2.Minimizeφ(X, rk) y using any of the unconstrained minimization methodsb
and obtain the solutionX∗k.
3 .Test whetherX∗kis the optimum solution of the original problem. IfX∗kis found
to be optimum, terminate the process. Otherwise, go to the next step.
4.Find the value of the next penalty parameter,rk+ 1 , as

rk+ 1 = crk

where c<1.

5.Set the new value ofk=k+1, take the new starting point asX 1 =X∗k, and

go to step 2.

Although the algorithm is straightforward, there are a number of points to be considered
in implementing the method:

1.The starting feasible pointX 1 may not be readily available in some cases.

2 .A suitable value of the initial penalty parameter (r 1 ) has to be found.

3 .A proper value has to be selected for the multiplication factor,c.