Engineering Optimization: Theory and Practice, Fourth Edition

7.15 Exterior Penalty Function Method 445

Figure 7.13 Aφfunction forq>1.

4.q>1. Theφfunction will have continuous first derivatives in this case as shown

in Fig. 7.13. These derivatives are given by

∂φ

∂xi

=

∂f

∂xi

+rk

∑m

j= 1

q〈gj(X)〉q−^1

∂gj(X)

∂xi

(7.202)

Generally, the value ofqis chosen as 2 in practical computation. We assume a

value ofq>1 in subsequent discussion of this method.

Algorithm. The exterior penalty function method can be stated by the following
steps:

1.Start from any designX 1 and a suitable value ofr 1. Set k= 1.

2.Find the vectorX∗kthat minimizes the function

φ(X, rk) =f(X)+rk

∑m

j= 1

〈gj(X)〉q

3. est whether the pointT X∗ksatisfies all the constraints. IfX∗kis feasible, it is the
   desired optimum and hence terminate the procedure. Otherwise, go to step 4.
   4.Choose the next value of the penalty parameter that satisfies the relation

rk+ 1 >rk

and set the new value ofkas originalkplus 1 and go to step 2. Usually,

the value ofrk+ 1 is chosen according to the relationrk+ 1 = crk, where cis a

constant greater than 1.