Engineering Optimization: Theory and Practice, Fourth Edition

436 Nonlinear Programming III: Constrained Optimization Techniques

1.The relative difference between the values of the objective function obtained

at the end of any two consecutive unconstrained minimizations falls below a

small numberε 1 , that is,

∣

∣

∣

∣

f(X∗k) −f(X∗k− 1 )

f (X∗k)

∣

∣

∣

∣≤ε^1 (7.167)

2 .The difference between the optimum pointsX∗kandX∗k− 1 becomes very small.

This can be judged in several ways. Some of them are given below:

|(  X)i| ≤ε 2 (7.168)

where X=X∗k−X∗k− 1 , and ( X)iis the ith component of the vector X.

max|(   X)i| ≤ε 3 (7.169)

| X|=[( X)^21 + ( X)^22 + · · · + ( X)^2 n]^1 /^2 ≤ε 4 (7.170)

Notethat the values ofε 1 toε 4 have to be chosen depending on the character-

istics of the problem at hand.

Normalization of Constraints. A structural optimization problem, for example, might

be having constraints on the deflection (δ) and the stress (σ) as

g 1 ( X)=δ(X)−δmax≤ 0 (7.171)

g 2 ( X)=σ(X)−σmax≤ 0 (7.172)

where the maximum allowable values are given by δmax= 0. 5 in. and σmax=

20 ,000 psi. If a design vectorX 1 gives the values ofg 1 andg 2 as− 0. 2 and− 10 ,000,

the contribution ofg 1 will be much larger than that of g 2 (by an order of 10^4 )

in the formulation of the φfunction given by Eq. (7.160). This will badly affect

the convergence rate during the minimization ofφfunction. Thus it is advisable to

normalize the constraints so that they vary between−1 and 0 as far as possible. For

the constraints shown in Eqs. (7.171) and (7.172), the normalization can be done as

g 1 ′(X)=

g 1 (X)

δmax

=

δ(X)

δmax

− 1 ≤ 0 (7.173)

g 2 ′(X)=

g 2 (X)

σmax

=

σ(X)

σmax

− 1 ≤ 0 (7.174)

If the constraints are not normalized as shown in Eqs. (7.173) and (7.174), the problem

can still be solved effectively by defining different penalty parameters for different

constraints as

φ(X, rk) =f(X)−rk

∑m

j= 1

Rj

gj(X)

(7.175)

whereR 1 , R 2 ,... , Rmare selected such that the contributions of differentgj( X)to the

φfunction will be approximately the same at the initial pointX 1. When the uncon-

strained minimization ofφ(X,rk) is carried for a decreasing sequence of values of

rk, the values ofR 1 , R 2 ,... , Rmwill not be altered; however, they are expected to be