Engineering Optimization: Theory and Practice, Fourth Edition

404 Nonlinear Programming III: Constrained Optimization Techniques

7.8 Rosen’s Gradient Projection Method

The gradient projection method of Rosen [7.9, 7.10] does not require the solution of an

auxiliary linear optimization problem to find the usable feasible direction. It uses the

projection of the negative of the objective function gradient onto the constraints that

are currently active. Although the method has been described by Rosen for a general

nonlinear programming problem, its effectiveness is confined primarily to problems in

which the constraints are all linear. Consider a problem with linear constraints:

Minimizef (X)

subject to

gj(X)=

∑n

i= 1

aijxi−bj≤ 0 , j= 1 , 2 ,... , m (7.50)

Let the indices of the active constraints at any point bej 1 , j 2 ,... , jp. The gradients of

the active constraints are given by

∇gj(X)=











a 1 j

a 2 j

..

.

anj











, j=j 1 , j 2 ,... , jp (7.51)

By defining a matrixNof ordern×pas

N=[∇gj 1 ∇gj 2.. .∇gjp] (7.52)

the direction-finding problem for obtaining a usable feasible directionScan be posed

as follows.

FindSwhich minimizesST∇ f(X) (7.53)

subject to

NTS= 0 (7.54)

STS− 1 = 0 (7.55)

where Eq. (7.55) denotes the normalization of the vector S. To solve this

equality-constrained problem, we construct the Lagrangian function as

L(S,λ, β)=ST∇ f(X)+λTNTS +β(STS− 1 ) (7.56)

where

λ=











λ 1

λ 2

..

.

λp









