Engineering Optimization: Theory and Practice, Fourth Edition

394 Nonlinear Programming III: Constrained Optimization Techniques

known asmethods of feasible directions. There are many ways of choosing usable

feasible directions, and hence there are many different methods of feasible directions.

As seen in Chapter 2, a directionSis feasible at a pointXiif it satisfies the relation

d

dλ

gj(Xi+λS)|λ= 0 =ST∇gj(Xi)≤ 0 (7.26)

where the equality sign holds true only if a constraint is linear or strictly concave,

as shown in Fig. 2.8. A vectorSwill be a usable feasible direction if it satisfies the

relations

d

dλ

f (Xi+λS)|λ= 0 =ST∇ f(Xi) < 0 (7.27)

d

dλ

gj(Xi+λS)|λ= 0 =ST∇gj(Xi)≤ 0 (7.28)

It is possible to reduce the value of the objective function at least by a small amount

by taking a step lengthλ>0 along such a direction.

The detailed iterative procedure of the methods of feasible directions will be con-

sidered in terms of two well-known methods: Zoutendijk’s method of feasible directions

and Rosen’s gradient projection method.

7.7 Zoutendijk’s Method of Feasible Directions

InZoutendijk’s method of feasible directions, the usable feasible direction is taken as

the negative of the gradient direction if the initial point of the iteration lies in the

interior (not on the boundary) of the feasible region. However, if the initial point

lies on the boundary of the feasible region, some constraints will be active and the

usable feasible direction is found so as to satisfy Eqs. (7.27) and (7.28). The iterative

procedure of Zoutendijk’s method can be stated as follows (only inequality constraints

are considered in Eq. (7.1), for simplicity.

Algorithm

1.Start with an initial feasible pointX 1 and small numbersε 1 ,ε 2 , andε 3 to test

theconvergence of the method. Evaluatef (X 1 ) nda gj(X 1 ) ,j= 1 , 2 ,... , m.

Set the iteration number asi=1.

2.Ifgj(Xi) < 0 ,j= 1 , 2 ,... , m(i.e.,Xi is an interior feasible point), set the

current search direction as

Si= −∇ f(Xi) (7.29)

NormalizeSiin a suitable manner and go to step 5. If at least onegj(Xi) = 0 ,

go to step 3.

3.Find a usable feasible directionSby solving the direction-finding problem:

Minimize−α (7.30a)