Engineering Optimization: Theory and Practice, Fourth Edition

446 Nonlinear Programming III: Constrained Optimization Techniques

Example 7.9

Minimizef (x 1 , x 2 )=^13 (x 1 + 1 )^3 +x 2

subject to

g 1 (x 1 , x 2 )= 1 −x 1 ≤ 0

g 2 (x 1 , x 2 ) =−x 2 ≤ 0

SOLUTION To illustrate the exterior penalty function method, we solve the uncon-

strained minimization problem by using differential calculus method. As such, it is not

necessary to have an initial trial pointX 1. The φfunction is

φ(X 1 , r)=^13 (x 1 + 1 )^3 +x 2 + maxr[ ( 0 , 1 −x 1 )]^2 + maxr[ ( 0 ,−x 2 )]^2

The necessary conditions for the unconstrained minimum ofφ(X, r)are

∂φ

∂x 1

= (x 1 + 1 )^2 − 2 r[max( 0 , 1 −x 1 )]= 0

∂φ

∂x 2

= 1 − 2 r[max( 0 ,−x 2 )]= 0

These equations can be written as

min[(x 1 + 1 )^2 , (x 1 + 1 )^2 − 2 r( 1 −x 1 )]= 0 (E 1 )

min[1, 1 + 2 rx 2 ]= 0 (E 2 )

In Eq. (E 1 ), if (x 1 + 1 )^2 = , 0 x 1 = − 1 (this violates the first constraint), and if

(x 1 + 1 )^2 − 2 r( 1 −x 1 ) = 0 , x 1 = − 1 −r+

√

r^2 + 4 r

In Eq. (E 2 ), the only possibility is that 1+ 2 rx 2 = and hence 0 x 2 = − 1 / 2 r.Thus the

solution of the unconstrained minimization problem is given by

x 1 ∗(r) =− 1 −r+r

(

1 +

4

r

) 1 / 2

(E 3 )

x 2 ∗(r) =−

1

2 r

(E 4 )

Fromthis, the solution of the original constrained problem can be obtained as

x 1 ∗= iml

r →∞

x∗ 1 (r) = 1 , x 2 ∗= iml

r →∞

x∗ 2 (r)= 0

fmin= iml

r →∞

φmin(r)=^83

The convergence of the method, asrincreases gradually, can be seen from Table 7.5.