Engineering Optimization: Theory and Practice, Fourth Edition

448 Nonlinear Programming III: Constrained Optimization Techniques

X∗ 1 ,X∗ 2 ,... ,X∗kconverges to the minimum pointX∗, and the sequencef 1 ∗, f 2 ∗,... , fk∗

to the minimum valuef∗of the original constrained problem stated in Eq. (7.153) as

rk→ 0.After carrying out a certain number of unconstrained minimizations ofφ, the

results obtained thus far can be used to estimate the minimum of the original constrained

problem by a method known as theextrapolation technique. The extrapolations of the

design vector and the objective function are considered in this section.

7.16.1 Extrapolation of the Design VectorX

Since different vectorsX∗i, i= 1 , 2 ,... , k, are obtained as unconstrained minima of

φ(X, ri) or differentf ri, i= 1 , 2 ,... , k, the unconstrained minimumφ(X, r)for any

value ofr,X∗(r) can be approximated by a polynomial in, ras

X∗(r)=

∑k−^1

j= 0

Aj(r)j=A 0 +rA 1 +r^2 A 2 + · · · +rk−^1 Ak− 1 (7.203)

whereAjare n-component vectors. By substituting the known conditions

X∗(r=ri)=X∗i, i= 1 , 2 ,... , k (7.204)

in Eq. (7.203), we can determine the vectorsAj,j= 0 , 1 , 2 ,... , k−1 uniquely. Then

X∗(r) given by Eq. (7.203), will be a good approximation for the unconstrained min-,

imum ofφ(X, r)in the interval( 0 , r 1 ) By setting. r=0 in Eq. (7.203), we can obtain

an estimate to the true minimum,X∗, as

X∗=X∗(r= 0 )=A 0 (7.205)

It is to be noted that it is not necessary to approximateX∗(r) by a (k−1) st-order

polynomial inr. In fact, any polynomial of order 1≤p≤k−1 can be used to approx-

imateX∗(r) In such a case we need only. p+1 points out ofX∗ 1 ,X∗ 2 ,... ,X∗kto define

the polynomial completely.

As a simplest case, let us consider approximatingX∗(r) y a first-order polynomialb

(linear equation) inras

X∗(r)=A 0 +rA 1 (7.206)

To evaluate the vectorsA 0 andA 1 , we need the data of two unconstrained minima. If

the extrapolation is being done at the end of thekth unconstrained minimization, we

generally use the latest information to find the constant vectorsA 0 andA 1. LetX∗k− 1

andX∗kbe the unconstrained minima corresponding tork− 1 andrk, respectively. Since

rk= crk− 1 ( <c 1), Eq. (7.206) gives

X∗(r=rk− 1 )=A 0 +rk− 1 A 1 =X∗k− 1

X∗(r=rk)=A 0 + crk− 1 A 1 =X∗k

(7.207)

These equations give

A 0 =

X∗k−cX∗k− 1

1 −c

A 1 =

X∗k− 1 −X∗k

rk− 1 ( 1 −c)

(7.208)