Engineering Optimization: Theory and Practice, Fourth Edition

7.7 Zoutendijk’s Method of Feasible Directions 399

Method 1. The optimal step length,λi, can be found by any of the one-dimensional
minimization methods described in Chapter 5. The only drawback with these methods
is that the constraints will not be considered while findingλi. Thus the new point
Xi+ 1 =Xi+λiSimay lie either in the interior of the feasible region (Fig. 7.8a), or on
the boundary of the feasible region (Fig. 7.8b), or in the infeasible region (Fig. 7.8c).
If the pointXi+ 1 lies in the interior of the feasible region, there are no active
constraints and hence we proceed to the next iteration by setting the new usable feasible
direction asSi+ 1 = −∇ f(Xi+ 1 ) (i.e.,we go to step 2 of the algorithm). On the other
hand, ifXi+ 1 lies on the boundary of the feasible region, we generate a new usable
feasible directionS=Si+ 1 by solving a new direction-finding problem (i.e., we go to
step 3 of the algorithm). One practical difficulty has to be noted at this stage. To detect
that pointXi+ 1 is lying on the constraint boundary, we have to find whether oneor
moregj(Xi+ 1 ) re zero. Since the computations are done numerically, will we say thata

Figure 7.8 Effect of taking optimal step length.