Engineering Optimization: Theory and Practice, Fourth Edition

392 Nonlinear Programming III: Constrained Optimization Techniques

subject to

− 2 ≤x 1 ≤ 2

− 2 ≤x 2 ≤ 2 (E 1 )

Thesolution of this problem can be obtained as

X=

[

− 2

2

]

withf (X)= − 4

Step 4: Since we have solved one LP problem, we can take

Xi+ 1 =X 2 =

{

− 2

2

}

Step 5:Sinceg 1 (X 2 ) = 23 >ε, we linearizeg 1 ( X)about pointX 2 as

g 1 (X)≃g 1 (X 2 ) +∇g 1 (X 2 )T(X−X 2 )≤ 0 (E 2 )

As

g 1 (X 2 ) = 23 ,

∂g 1

∂x 1

∣

∣

∣

∣

X 2

=( 6 x 1 − 2 x 2 )|X 2 =− 16

∂g 1

∂x 2

∣

∣

∣

∣

X 2

=(− 2 x 1 + 2 x 2 )|X 2 = 8

Eq. (E 2 ) becomes

g 1 ( X)≃− 16 x 1 + 8 x 2 − 52 ≤ 0

By adding this constraint to the previous LP problem, the new LP prob-

lem becomes

Minimizef=x 1 −x 2

subject to

− 2 ≤x 1 ≤ 2

− 2 ≤x 2 ≤ 2 (E 3 )

− 16 x 1 + 8 x 2 − 52 ≤ 0

Step 6: Set the iteration number asi=2 and go to step 4.

Step 4: Solve the approximating LP problem stated in Eqs. (E 3 ) and obtain the

solution

X 3 =

{

− 0. 5625

2. 0

}

withf 3 = f(X 3 ) =− 2. 5625

This procedure is continued until the specified convergence criterion,

g 1 (Xi) ≤ε,in step 5 is satisfied. The computational results are summa-

rized in Table 7.2.