7.7 Zoutendijk’s Method of Feasible Directions 395
subject to
ST∇gj(Xi)+θjα ≤ 0 , j= 1 , 2 ,... , p (7.30b)
ST∇f+α≤ 0 (7.30c)
− 1 ≤si≤ 1 , i= 1 , 2 ,... , n (7.30d)
wheresiis the ith component ofS, the firstpconstraints have been assumed
to be active at the pointXi(the constraints can always be renumbered to satisfy
this requirement), and the values of allθjcan be taken as unity. Hereαcan be
taken as an additional design variable.
4.If the value ofα∗found in step 3 is very nearly equal to zero, that is, ifα∗≤ε 1 ,
terminate the computation by takingXopt≃Xi. Ifα∗>ε 1 , go to step 5 by taking
Si=S.
5 .Find a suitable step lengthλialong the directionSi and obtain a new point
Xi+ 1 as
Xi+ 1 =Xi+λiSi (7.31)
The methods of finding the step lengthλiwill be considered later.
6 .Evaluate the objective functionf (Xi+ 1 ).
7 .Test for the convergence of the method. If
∣
∣
∣
∣
f (Xi) −f(Xi+ 1 )
f (Xi)
∣
∣
∣
∣≤ε^2 and^ ||Xi−Xi+^1 || ≤ε^3 (7.32)
terminate the iteration by takingXopt≃Xi+ 1. Otherwise, go to step 8.
8 .Set the new iteration number asi=i+1, and repeat from step 2 onward.
There are several points to be considered in applying this algorithm. These are
related to (1) finding an appropriate usable feasible direction (S), (2) finding a suitable
step size along the directionS, and (3) speeding up the convergence of the process.
All these aspects are discussed below.
7.7.1 Direction-Finding Problem
If the pointXi lies in the interior of the feasible region [i.e.,gj(Xi) < 0 forj=
1 , 2 ,... , m], the usable feasible direction is taken as
Si= −∇ f(Xi) (7.33)
The problem becomes complicated if one or more of the constraints are critically
satisfied atXi, that is, when some of thegj(Xi) = 0. One simple way to find a usable
feasible direction at a pointXiat which some of the constraints are active is to generate
arandom vector and verify whether it satisfies Eqs. (7.27) and (7.28). This approach
is a crude one but is very simple and easy to program. The relations to be checked for
each random vector are also simple, and hence it will not require much computer time.
However, a more systematic procedure is generally adopted to find a usable feasible
direction in practice. Since there will be, in general, several directions that satisfy
