Engineering Optimization: Theory and Practice, Fourth Edition

7.8 Rosen’s Gradient Projection Method 409

value of

df

dλ

=STi∇ f(λ) at λ=λM

If the minimum value ofλ,λ∗i, lies in between λ= 0 andλ=λM, the quantity
df/dλ(λM) ill be positive. In such a case we can find the minimizing step lengthw λ∗i
by interpolation or by using any of the techniques discussed in Chapter 5.
An important point to be noted is that if the step length is given byλi(not by
λ∗i), at least one more constraint will be active atXi+ 1 than atXi. These additional
constraints will have to be considered in generating the projection matrix atXi+ 1. On
the other hand, if the step length is given byλ∗i, no new constraint will be active at
Xi+ 1 , and hence the projection matrix atXi+ 1 involves only those constraints that were
active atXi.

Algorithm. The procedure involved in the application of the gradient projection
method can be described by the following steps:

1.Start with an initial pointX 1. The pointX 1 has to be feasible, that is,

gj(X 1 ) ≤ 0 , j= 1 , 2 ,... , m

2.Set the iteration number asi=1.

3.IfXiis an interior feasible point [i.e., ifgj(Xi) < 0 forj= 1 , 2 ,... , m], set

the direction of search asSi= −∇ f(Xi) normalize the search direction as,

Si=

−∇f (Xi)

||∇f (Xi) ||

and go to step 5. However, ifgj(Xi) = 0 forj=j 1 , j 2 ,... , jp, go to step 4.

4 .Calculate the projection matrixPias

Pi=I−Np(NTpNp)−^1 NTp

where

Np=[∇gj 1 (Xi)∇gj 2 (Xi).. .∇gjp(Xi)]

and find the normalized search directionSias

Si=

−Pi∇ f(Xi)

||Pi∇ f(Xi) ||

5.Test whether or notSi= 0 .IfSi= 0 ,go to step 6. IfSi= 0 ,compute the

vectorλatXias

λ=−(NTpNp)−^1 NTp∇ f(Xi)

If all the components of the vectorλare nonnegative, takeXopt=Xiand stop

the iterative procedure. If some of the components ofλare negative, find the

componentλqthat has the most negative value and form the new matrixNpas

Np=[∇gj 1 ∇gj 2 · · · ∇gj q− 1 ∇gj q+ 1 · · · ∇gjp]

and go to step 3.