Engineering Optimization: Theory and Practice, Fourth Edition

464 Nonlinear Programming III: Constrained Optimization Techniques

Table 7.6 Results for Example 7.12

λ(i) rk x 1 ∗(i) x∗ 2 (i) Value ofh

0.00000 1.00000 −0.23810 2.22222 −3.01587

−6.03175 1.00000 −0.38171 3.56261 −1.81910

−9.66994 1.00000 −0.46833 4.37110 −1.09723

−11.86441 1.00000 −0.52058 4.85876 −0.66182

−13.18806 1.00000 −0.55210 5.15290 −0.39919

−13.98645 1.00000 −0.57111 5.33032 −0.24078

−14.46801 1.00000 −0.58257 5.43734 −0.14524

−14.75848 1.00000 −0.58949 5.50189 −0.08760

−14.93369 1.00000 −0.59366 5.54082 −0.05284

−15.03937 1.00000 −0.59618 5.56430 −0.03187

For the stationary point ofA, the necessary conditions,∂A/∂xi= , 0 i=1, 2, yield

x 1 2 ( 1 + 2 rk)+x 2 ( 4 + 2 rk) = 10 rk−λ (E 4 )

x 1 ( 4 + 2 rk)+x 2 ( 6 + 2 rk) = 10 rk−λ (E 5 )

The solution of Eqs. (E 4 ) nd (Ea 5 ) ivesg

x 1 =

− 90 rk^2 + 9 rkλ − 6 λ+ 60 rk

( 14 − 5 rk)( 21 + 2 rk)

(E 6 )

x 2 =

20 rk− 2 λ

14 − 5 rk

(E 7 )

Let the value ofrkbe fixed at 1 and select a value ofλ(^1 )=. This gives 0

x 1 ∗(^1 )= − 215 , x 2 ∗(^1 )=^209 with h=− 215 +^209 − 5 =− 3. 01587

For the next iteration,

λ(^2 )=λ(^1 )+ 2 rkh(X∗(^1 ))= 0 + 2 ( 1 )(− 3. 01587 )= − 6. 03175

Substituting this value forλalong withrk= in Eqs. (E 1 6 ) nd (Ea 7 ) we get,

x 1 ∗(^2 )= − 0. 38171 , x∗ 2 (^2 )= 3. 56261

with h= − 0. 38171 + 3. 56261 − 5 = − 1. 81910

This procedure can be continued until some specified convergence is satisfied. The

results of the first ten iterations are given in Table 7.6.

7.21 CHECKING THE CONVERGENCE OF CONSTRAINED
OPTIMIZATION PROBLEMS
In all the constrained optimization techniques described in this chapter, identification
of the optimum solution is very important from the points of view of stopping the