Engineering Optimization: Theory and Practice, Fourth Edition

684 Optimal Control and Optimality Criteria Methods

determined by other considerations). Assuming that the firstnvariables denote the

active variables, we can rewrite Eqs. (12.46) and (12.47) as

f=f+

∑n

i= 1

cixi (12.52)

∑n

i= 1

ai

xi

=ymax−y=y∗ (12.53)

wherefandydenote the contribution of the passive variables tofandy, respectively.

Equation (12.51) now gives

xk=

√

λ

√

ak

ck

, k= 1 , 2 ,... ,n (12.54)

Substituting Eq. (12.54) into Eq. (12.53), and solving forλ, we obtain

√

λ=

1

y∗

∑n

k= 1

√

akck (12.55)

Using Eq. (12.55) in Eq. (12.54) results in

xk=

1

y∗

√

ak

ck

∑n

i= 1

√

aici, k= 1 , 2 ,... ,n (12.56)

Equation (12.56) is the optimality criteria that must be satisfied at the optimum solu-

tion of the problem stated by Eqs. (12.46) and (12.47). This equation can be used to

iteratively update the design variablesxkas

xk(j+^1 )=

(

1

y∗

√

ak

ck

∑n

i= 1

√

aici

)(j )

, k= 1 , 2 ,... ,n (12.57)

where the superscriptjdenotes the iteration cycle. In each iteration, the components

akandckare assumed to be constants (in general, they depend on the design vector).

12.4.2 Optimality Criteria with Multiple Displacement Constraints

When multiple displacement constraints are included, as in the case of a structure sub-

jected to multiple-load conditions, the optimization problem can be stated as follows:

Find a set of active variablesX= {x 1 x 2... xn}Twhich minimizes

f(X)=f 0 +

∑n

i= 1

cixi (12.58)

subjectto

yj=

∑n

i= 1

aj i

xi

=yj∗, j= 1 , 2 ,... , J (12.59)