Engineering Optimization: Theory and Practice, Fourth Edition

7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints 453

Figure 7.14 Linear extended penalty function method.

Eq. (7.225) compared to Eq. (7.222). The concept of extended interior penalty function

approach can be generalized to define a variable penalty function method from which

the linear and quadratic methods can be derived as special cases [7.24].

Example 7.11 Plot the contours of theφkfunction using the linear extended interior

penalty function for the following problem:

Minimizef (x)=(x− 1 )^2

subject to

g 1 (x)= 2 −x≤ 0

g 2 (x)=x− 4 ≤ 0

SOLUTION We choosec= 0 .2 anda= 0 .5 so thatε= − 0. 2

√

rk. Theφkfunction

is defined by Eq. (7.222). By selecting the values ofrk as 10.0, 1.0, 0.1, and 0.01

sequentially, we can determine the values ofφk for different values ofx,which can

then be plotted as shown in Fig. 7.15. The graph off (x)is also shown in Fig. 7.15

for comparison.

7.18 PENALTY FUNCTION METHOD FOR PROBLEMS
WITH MIXED EQUALITY AND INEQUALITY CONSTRAINTS
The algorithms described in previous sections cannot be directly applied to solve prob-
lems involving strict equality constraints. In this section we consider some of the