Engineering Optimization: Theory and Practice, Fourth Edition

450 Nonlinear Programming III: Constrained Optimization Techniques

7.16.2 Extrapolation of the Functionf

As in the case of the design vector, it is possible to use extrapolation technique

to estimate the optimum value of the original objective function,f∗. For this, let

f 1 ∗, f 2 ∗,... , fk∗be the values of the objective function corresponding to the vectors

X∗ 1 ,X∗ 2 ,... ,X∗k. Since the pointsX∗ 1 ,X∗ 2 ,... ,X∗khave been found to be the uncon-

strained minima of theφ function corresponding tor 1 , r 2 ,... , rk, respectively, the

objective function,f∗, can be assumed to be a function ofr.By approximatingf∗by

a(k−1)st-order polynomial inr, we have

f∗(r)=

k∑− 1

j= 0

aj(r)j=a 0 +a 1 r+a 2 r^2 + · · · +ak− 1 rk−^1 (7.213)

where thekconstantsaj,j= 0 , 1 , 2 ,... , k−1 can be evaluated by substituting the

known conditions

f∗(r=ri)=fi∗=a 0 +a 1 ri+a 2 r^2 i+ · · · +ak− 1 rik−^1 , i= 1 , 2 ,... , k (7.214)

Since Eq. (7.213) is a good approximation for the truef∗in the interval( 0 ,r 1 ) we,

can obtain an estimate for the constrained minimum off as

f∗≃f∗(r= 0 )=a 0 (7.215)

As a particular case, a linear approximation can be made forf∗by using the last two

data points. Thus iffk∗− 1 andfk∗are the function values corresponding tork− 1 and

rk= crk− 1 , we have

fk∗− 1 =a 0 +rk− 1 a 1

fk∗=a 0 + crk− 1 a 1

(7.216)

These equations yield

a 0 =

fk∗− cfk∗− 1

1 −c

(7.217)

a 1 =

fk∗− 1 −fk∗

rk− 1 ( 1 −c)

(7.218)

f∗(r)=

fk∗− cfk∗− 1

1 −c

+

r

rk− 1

fk∗− 1 −fk∗

1 −c

(7.219)

Equation (7.219) gives an estimate off∗as

f∗≃f∗(r= 0 )=a 0 =

fk∗− cfk∗− 1

1 −c

(7.220)

The extrapolated valuea 0 can be used to provide an additional convergence criterion

for terminating the interior penalty function method. The criterion is that whenever the

value offk∗obtained at the end ofkth unconstrained minimization ofφis sufficiently

close to the extrapolated valuea 0 , that is, when

∣

∣

∣

∣

fk∗−a 0

fk∗

∣

∣

∣

∣≤ε (7.221)

whereεis a specified small quantity, the process can be terminated.