Engineering Optimization: Theory and Practice, Fourth Edition

456 Nonlinear Programming III: Constrained Optimization Techniques

As in the case of Eq. (7.199), this function has to be minimized for an increasing

sequence of values ofrk. It can be proved that asrk→ ∞ ,the unconstrained minima,

X∗k, of φ(X, rk) onverge to the minimum of the original constrained problem statedc

in Eq. (7.227).

7.19 Penalty Function Method for Parametric Constraints

7.19.1 Parametric Constraint

In some optimization problems, a particular constraint may have to be satisfied over a

range of some parameter (θ) as

gj( X,θ )≤ 0 , θl≤θ≤θu (7.231)

whereθlandθuare lower and the upper limits onθ,respectively. These types of

constraints are calledparametric constraints. As an example, consider the design of

a four-bar linkage shown in Fig. 7.16. The angular position of the output linkφwill

depend on the angular position of the input link,θ, and the lengths of the links,l 1 ,l 2 ,

l 3 , andl 4. Ifli(i = 1 to 4)are taken as the design variablesxi(i = 1 to 4), the angular

position of the output link,φ(X, θ ), for any fixed value ofθ (θi) an be changed byc

changing the design vector,X. Thus ifφ(θ )is the output desired, the outputφ(X, θ )

generated will, in general, be different from that ofφ(θ ), as shown in Fig. 7.17. If the

linkage is used in some precision equipment, we would like to restrict the difference

|φ(θ )−φ(X, θ )|to be smaller than some permissible value, say,ε. Since this restriction

has to be satisfied for all values of the parameterθ, the constraint can be stated as a

parametric constraint as

|φ(θ )−φ(X, θ )| ≤ε, 0 ◦≤ θ≤ 360 ◦ (7.232)

Sometimes the number of parameters in a parametric constraint may be more than

one. For example, consider the design of a rectangular plate acted on by an arbitrary load

as shown in Fig. 7.18. If the magnitude of the stress induced under the given loading,

|σ (x, y)|, is restricted to be smaller than the allowable valueσmax, the constraint can

Figure 7.16 Four-bar linkage.