7.17 Extended Interior Penalty Function Methods 451
Example 7.10 Find the extrapolated values ofXandf in Example 7.8 using the
results of minimization ofφ(X, r 1 ) nda φ(X, r 2 ).
SOLUTION From the results of Example 7.8, we have forr 1 = 1. 0 ,
X∗ 1 =
0. 37898
1. 67965
2. 34617
, f 1 ∗= 5. 70766
and forr 2 = 0. 1 ,
c= 0. 1 , X∗ 2 =
0. 10088
1. 41945
1. 68302
, f 2 ∗= 2. 73267
By using Eq. (7.206) for approximatingX∗(r) the extrapolated vector, X∗is given by
Eq. (7.209) as
X∗≃A 0 =
X∗ 2 −cX∗ 1
1 −c
=
1
0. 9
0. 10088
1. 41945
1. 68302
− 0. 1
0. 37898
1. 67865
2. 34617
(E 1 )
=
0. 06998
1. 39053
1. 60933
(E 2 )
Similarly, the linear resltionshipsf∗(r)=a 0 +a 1 r eads to [from Eq. (7.220)]l
f∗≃
f 2 ∗− cf 1 ∗
1 −c
=
1
0. 9
[2. 73267 − 0. 1 ( 5. 707667 )]= 2. 40211 (E 3 )
It can be verified that the extrapolated design vectorX∗is feasible and hence can be
used as a better starting point for the subsequent minimization of the functionφ.
7.17 Extended Interior Penalty Function Methods
In the interior penalty function approach, theφfunction is defined within the feasible
domain. As such, if any of the one-dimensional minimization methods discussed in
Chapter 5 is used, the resulting optimal step lengths might lead to infeasible designs.
Thus the one-dimensional minimization methods have to be modified to avoid this prob-
lem. An alternative method, known as theextended interior penalty function method,
has been proposed in which theφfunction is defined outside the feasible region. The
extended interior penalty function method combines the best features of the interior and
exterior methods for inequality constraints. Several types of extended interior penalty
function formulations are described in this section.
7.17.1 Linear Extended Penalty Function Method
The linear extended penalty function method was originally proposed by Kavlie and
Moe [7.18] and later improved by Cassis and Schmit [7.19]. In this method, theφk
