Engineering Optimization: Theory and Practice, Fourth Edition

10.8 Generalized Penalty Function Method 621

Figure 10.10 Solution of a single-variable integer problem by penalty function method.x 1 ,
discrete variable;xj 1 , jth value ofx 1 [10.4].

Choice of the Initial Values ofrk,sk, andβk. The numerical values ofrk,sk, and
βkhave to be chosen carefully to achieve fast convergence. If these values are chosen
such that they give the response surfaces ofφfunction as shown in Fig. 10.10c, several
local minima will be introduced and the risk in finding the global minimum point will
be more. Hence the initial value ofsk(namely,s 1 ) is to be chosen sufficiently small
to yield a unimodal response surface. This can be achieved by setting

skQ′k≪Pk′ (10.78)

whereQ′kis an estimate of the maximum magnitude of the gradient to theQksurface
andPk′is a measure of the gradient of the functionPkdefined by

Pk= f(X)+rk

∑m

j= 1

Gj[gj(X)] (10.79)

Gisvold and Moe [10.4] have taken the values ofQ′kandPk′as

Q′k=^12 · 4 βkβk(βk− 1 )βk−^1 ( 2 βk− 1 )^1 /^2 −βk (10.80)

Pk′=

(

∇PkT∇Pk

n

) 1 / 2

(10.81)