7.16 Extrapolation Techniques in the Interior Penalty Function Method 447Table 7.5 Results for Example 7.9
Value ofr x∗ 1 x∗ 2 φmin(r) fmin(r)
0.001 − 0. 93775 − 500. 00000 − 249. 9962 − 500. 0000
0.01 − 0. 80975 − 50. 00000 − 24. 9650 − 49. 9977
0.1 − 0. 45969 − 5. 00000 − 2. 2344 − 4. 9474
1 0.23607 − 0. 50000 0.9631 0.1295
10 0.83216 − 0. 05000 2.3068 2.0001
100 0.98039 − 0. 00500 2.6249 2.5840
1,000 0.99800 − 0. 00050 2.6624 2.6582
10,000 0.99963 − 0. 00005 2.6655 2.6652∞ (^108383)
Convergence Proof. To prove the convergence of the algorithm given above, we
assume thatf andgj, j= 1 , 2 ,... , m, are continuous and that an optimum solution
exists for the given problem. The following results are useful in proving the convergence
of the exterior penalty function method.
Theorem 7.3If
φ(X, rk) =f(X)+rkG[g(X)]=f(X)+rk
∑m
j= 1
〈gj(X)〉q
the following relations will be valid for any 0<rk< rk+ 1 :
1 .φ(X∗k, rk) ≤φ(X∗k+ 1 , rk+ 1 ).
2 .f (X∗k) ≤f(X∗k+ 1 ).
3 .G[g(X∗k)]≥G[g(X∗k+ 1 ) .]
Proof: The proof is similar to that of Theorem 7.1.
Theorem 7.4If the functionφ(X, rk) iven by Eq. (7.199) is minimized for an increas-g
ing sequence of values ofrk, the unconstrained minimaX∗kconverge to the optimum
solution (X∗) of the constrained problem asrk→ ∞.
Proof: The proof is similar to that of Theorem 7.1 (see Problem 7.46).
7.16 Extrapolation Techniques in the Interior Penalty Function Method
In the interior penalty function method, theφfunction is minimized sequentially for
a decreasing sequence of valuesr 1 >r 2 >·· ·>rkto find the unconstrained minima
X∗ 1 ,X∗ 2 ,... ,X∗k, respectively. Let the values of the objective function corresponding to
X∗ 1 ,X∗ 2 ,... ,X∗kbef 1 ∗, f 2 ∗,... , fk∗, respectively. It has been proved that the sequence