Engineering Optimization: Theory and Practice, Fourth Edition

460 Nonlinear Programming III: Constrained Optimization Techniques

subject to

hj( X)= 0 , j= 1 , 2 ,... , p, p < n (7.240)

The Lagrangian corresponding to Eqs. (7.239) and (7.240) is given by

L(X,λ)=f (X)+

∑p

j= 1

λjhj(X) (7.241)

whereλj, j= 1 , 2 ,... , p, are the Lagrange multipliers. The necessary conditions for

a stationary point ofL(X,λ)include the equality constraints, Eq. (7.240). The exterior

penalty function approach is used to define the new objective functionA(X,λ, rk),

termed theaugmented Lagrangian function, as

A(X,λ, rk) =f(X)+

∑p

j= 1

λjhj(X)+rk

∑p

j= 1

h^2 j(X) (7.242)

whererkis the penalty parameter. It can be noted that the functionAreduces to the

Lagrangian ifrk= and to the 0 φfunction used in the classical penalty function method

if allλj=. It can be shown that if the Lagrange multipliers are fixed at their optimum 0

valuesλ∗j, the minimization ofA(X,λ,rk) ives the solution of the problem stated ing

Eqs. (7.239) and (7.240) in one step for any value ofrk. In such a case there is no need

to minimize the functionAfor an increasing sequence of values ofrk. Since the values

ofλ∗jare not known in advance, an iterative scheme is used to find the solution of the

problem. In the first iteration (k=1), the values ofλ(k)j are chosen as zero, the value

ofrkis set equal to an arbitrary constant, and the functionAisminimized with respect

toXto findX∗ k)(. The values ofλ(k)j andrkare then updated to start the next iteration.

For this, the necessary conditions for the stationary point ofL, given by Eq. (7.241),

are written as

∂L

∂xi

=

∂f

∂xi

+

∑p

j= 1

λ∗j

∂hj

∂xi

= 0 , i= 1 , 2 ,... , n (7.243)

whereλ∗jdenote the values of Lagrange multipliers at the stationary point ofL.Simi-

larly, the necessary conditions for the minimum ofAcan be expressed as

∂A

∂xi

=

∂f

∂xi

+

∑p

j= 1

(λj+ 2 rkhj)

∂hj

∂xi

= 0 , i= 1 , 2 ,... , n (7.244)

A comparison of the right-hand sides of Eqs. (7.243) and (7.244) yields

λ∗j=λj+ 2 rkhj, j= 1 , 2 ,... , p (7.245)

These equations are used to update the values ofλjas

λ(kj+^1 )=λ(k)j + 2 rkhj(X(k)), j= 1 , 2 ,... , p (7.246)