Engineering Optimization: Theory and Practice, Fourth Edition

438 Nonlinear Programming III: Constrained Optimization Techniques

Table 7.3 Results for Example 7.7

Value ofr x 1 ∗(r)=(r^1 /^2 + 1 )^1 /^2 x∗ 2 (r)=r^1 /^2 φmin(r) f (r)

1000 5.71164 31.62278 376.2636 132.4003

100 3.31662 10.00000 89.9772 36.8109

10 2.04017 3.16228 25.3048 12.5286

1 1.41421 1.00000 9.1046 5.6904

0.1 1.14727 0.31623 4.6117 3.6164

0.01 1.04881 0.10000 3.2716 2.9667

0.001 1.01569 0.03162 2.8569 2.7615

0.0001 1.00499 0.01000 2.7267 2.6967

0.00001 1.00158 0.00316 2.6856 2.6762

0.000001 1.00050 0.00100 2.6727 2.6697

Exact solution 0 1 0 8/3 8/3

SOLUTION The interior penalty function method, coupled with the Davidon–Fletcher

–Powell method of unconstrained minimization and cubic interpolation method of

one-dimensional search, is used to solve this problem. The necessary data are assumed

as follows:

Starting feasible point, X 1 =







0. 1

0. 1

3. 0







r 1 = 1. 0 , f (X 1 ) = 4. 041 , φ(X 1 , r 1 ) = 25. 1849

The optimum solution of this problem is known to be [7.15]

X=







0

√

2

√

2







, f∗=

√

2

The results of numerical optimization are summarized in Table 7.4.

Convergence Proof. The following theorem proves the convergence of the interior

penalty function method.

Theorem 7.1If the function

φ(X, rk) =f(X)−rk

∑m

j= 1

1

gj(X)

(7.176)

is minimized for a decreasing sequence of values ofrk, the unconstrained minimaX∗k

converge to the optimal solution of the constrained problem stated in Eq. (7.153) as

rk→ 0.