Engineering Optimization: Theory and Practice, Fourth Edition

462 Nonlinear Programming III: Constrained Optimization Techniques

7.20.2 Inequality-Constrained Problems

Consider the following inequality-constrained problem:

Minimizef (X) (7.248)

subject to

gj( X)≤ 0 , j= 1 , 2 ,... , m (7.249)

To apply the ALM method, the inequality constraints of Eq. (7.249) are first converted

to equality constraints as

gj(X)+yj^2 = 0 , j= 1 , 2 ,... , m (7.250)

whereyj^2 are the slack variables. Then the augmented Lagrangian function is con-

structed as

A(X,λ,Y, rk) =f(X)+

∑m

j= 1

λj[gj(X)+y^2 j]+

∑m

j= 1

rk[gj(X)+y^2 j]^2 (7.251)

where the vector of slack variables,Y, is given by

Y=











y 1

y 2

..

.

ym











If the slack variablesyj, j= 1 , 2 ,... , m, are considered as additional unknowns, the

functionAis to be minimized with respect toXandYfor specified values ofλjand

rk. This increases the problem size. It can be shown [7.23] that the functionAgiven

by Eq. (7.251) is equivalent to

A(X,λ, rk) =f(X)+

∑m

j= 1

λjαj+rk

∑m

j= 1

αj^2 (7.252)

where

αj= axm

{

gj( X),−

λj

2 rk

}

(7.253)

Thus the solution of the problem stated in Eqs. (7.248) and (7.249) can be obtained by

minimizing the functionA, given by Eq. (7.252), as in the case of equality-constrained

problems using the update formula

λ(kj+^1 )=λ(k)j + 2 rkα(k)j , j= 1 , 2 ,... , m (7.254)

in place of Eq. (7.246). It is to be noted that the functionA, given by Eq. (7.252), is

continuous and has continuous first derivatives but has discontinuous second derivatives

with respect toXatgj( X)=−λj/ 2 rk. Hence a second-order method cannot be used

to minimize the functionA.