Engineering Optimization: Theory and Practice, Fourth Edition

256 Nonlinear Programming I: One-Dimensional Minimization Methods

i Value ofs xi=x 1 +s fi=f(xi) Isfi>fi− 1?

1 — 0.0 0.0 —

2 0.05 0.05 −0.0725 No

3 0.10 0.10 −0.140 No

4 0.20 0.20 −0.260 No

5 0.40 0.40 −0.440 No

6 0.80 0.80 −0.560 No

7 1.60 1.60 + 0. 160 Yes

From these results, the optimum point can be seen to bexopt≈x 6 = 0. 8. In this case,

the pointsx 6 andx 7 do not really bracket the minimum point but provide informati on

about it. If a better approximation to the minimum is desired, the procedure can be

restarted fromx 5 with a smaller step size.

5.4 Exhaustive Search

The exhaustive search method can be used to solve problems where the interval in

which the optimum is known to lie is finite. Letxsandxfdenote, respectively, the

startingand final points of the interval of uncertainty.†The exhaustive search method

consists of evaluating the objective function at a predetermined number of equally

spaced points in the interval (xs, xf) and reducing the interval of uncertainty using the,

assumption of unimodality. Suppose that a function is defined on the interval (xs, xf)

and let it be evaluated at eight equally spaced interior pointsx 1 tox 8. Assuming that

the function values appear as shown in Fig. 5.6, the minimum point must lie, according

to the assumption of unimodality, between pointsx 5 andx 7. Thus the interval (x 5 , x 7 )

can be considered as the final interval of uncertainty.

In general, if the function is evaluated atnequally spaced points in the original

interval of uncertainty of lengthL 0 =xf−xs, and if the optimum value of the function

(among thenfunction values) turns out to be at pointxj, the final interval of uncertainty

Figure 5.6 Exhaustive search.

†Since the interval (xs, xf) but not the exact location of the optimum in this interval, is known to us, the,

interval (xs, xf) s called thei interval of uncertainty.