Engineering Optimization: Theory and Practice, Fourth Edition

754 Practical Aspects of Optimization

Similarly, a currently inactive constraint will become critical due topif the new

value ofgjbecomes zero:

gj(X)+

dgj

dp

p=gj(X)+

(n

∑

i= 1

∂gj

∂xi

∂xi

∂p

)

p (14.80)

Thus the change p necessary to make an inactive constraint active can be

found as

p= −

gj(X)

∑n

i= 1

∂gj

∂xi

∂xi

∂p

(14.81)

1 4.7.2 Sensitivity Equations Using the Concept of Feasible Direction

Here we treat the problem parameterpas a design variable so that the new design

vector becomes

X= {x 1 x 2 · · ·xnp}T (14.82)

As in the case of the method of feasible directions (see Section 7.7), we formulate the

direction finding problem as

FindXwhich minimizes−ST∇ f(X)

subject to

ST∇gj≤ 0 , j∈J 1

STS≤ 1 (14.83)

where the gradients offandgj(j∈J 1 ) can be evaluated in the usual manner. The set

J 1 can include nearly active constraints also (along with the active constraints) so that

we do not violate any constraint due to the change,p. The solution of the problem

stated in Eqs. (14.83) gives a usable feasible search direction,S. A new design vector

alongScan be expressed as

Xnew=Xcurrent+λS=Xcurrent+X (14.84)

whereλis the step length and the components ofScan be considered as

si=











∂xi

∂λ

, i= 1 , 2 ,... , n

∂p

∂λ

, i=n+ 1

(14.85)

so that

p=λsn+ 1 or λ=

p

sn+ 1

(14.86)