Engineering Optimization: Theory and Practice, Fourth Edition

418 Nonlinear Programming III: Constrained Optimization Techniques

If the vectorXnewcorresponding toλ∗is found infeasible, thenYnewis held

constant andZnewis modified using Eq. (7.108) withdZ=Znew−Zold.

Finally, when convergence is achieved with Eq. (7.108), we find that

Xnew=

{

Yold+ Y

Zold+ Z

}

(7.116)

and go to step 1.

Example 7.4

Minimizef (x 1 , x 2 , x 3 )=(x 1 −x 2 )^2 + (x 2 −x 3 )^4

subject to

g 1 (X)=x 1 ( 1 +x^22 )+x 34 − 3 = 0

− 3 ≤xi≤ 3 , i= 1 , 2 , 3

using the GRG method.

SOLUTION

Step 1: We choose arbitrarily the independent and dependent variables as

Y=

{

y 1

y 2

}

=

{

x 1

x 2

}

, Z={z 1 } = {x 3 }

Let the starting vector be

X 1 =







− 2. 6

2

2







withf (X 1 ) = 21 .16.

Step 2: Compute the GRG atX 1. Noting that

∂f

∂x 1

= 2 (x 1 −x 2 )

∂f

∂x 2

= − 2 (x 1 −x 2 ) + 4 (x 2 −x 3 )^3

∂f

∂x 3

= − 4 (x 2 −x 3 )^3

∂g 1

∂x 1

= 1 +x^22

∂g 1

∂x 2

= 2 x 1 x 2

∂g 1

∂x 3

= 4 x 33