Engineering Optimization: Theory and Practice, Fourth Edition

452 Nonlinear Programming III: Constrained Optimization Techniques

function is constructed as follows:

φk= φ(X, rk) =f(X)+rk

∑m

j= 1

g ̃j(X) (7.222)

where

g ̃j(X)=











−

1

gj(X)

ifgj(X)≤ε

−

2 ε−gj(X)

ε^2

ifgj(X)>ε

(7.223)

andεis a small negative number that marks the transition from the interior penalty

[gj( X)≤ε]to the extended penalty [gj( X)>ε]. To produce a sequence of improved

feasible designs, the value ofεis to be selected such that the functionφkwill have a

positive slope at the constraint boundary. Usually,εis chosen as

ε= −c(rk)a (7.224)

wherecandaare constants. The constantais chosen such that^13 ≤a≤^12 , where

the value ofa=^13 guarantees that the penalty for violating the constraints increases

asrk goes to zero while the value ofa=^12 is required to help keep the minimum

pointX∗in the quadratic range of the penalty function. At the start of optimization,

εis selected in the range− 0. 3 ≤ε≤ − 0 .1. The value ofr 1 is selected such that the

values off (X)andr 1

∑m

j= 1 g ̃j(^ X)areequal at the initial design vectorX^1. This defines

the value ofcin Eq. (7.224). The value ofεis computed at the beginning of each

unconstrained minimization using the current value ofrkfrom Eq. (7.224) and is kept

constant throughout that unconstrained minimization. A flowchart for implementing the

linear extended penalty function method is given in Fig. 7.14.

7.17.2 Quadratic Extended Penalty Function Method

Theφkfunction defined by Eq. (7.222) can be seen to be continuous with continuous

first derivatives atgj( X)=ε.However, the second derivatives can be seen to be

discontinuous atgj( X)=ε.Hence it is not possible to use a second-order method for

unconstrained minimization [7.20]. The quadratic extended penalty function is defined

so as to have continuous second derivatives atgj( X)=εas follows:

φk= φ(X, rk) =f(X)+rk

∑m

j= 1

g ̃j(X) (7.225)

where

g ̃j(X)=















−

1

gj(X)

if gj(X)≤ε

{

−

1

ε

[

gj(X)

ε

] 2

− 3

gj(X)

ε

+ 3

}

if gj(X)>ε

(7.226)

With this definition, second-order methods can be used for the unconstrained mini-

mization ofφk. It is to be noted that the degree of nonlinearity ofφkis increased in