Engineering Optimization: Theory and Practice, Fourth Edition

314 Nonlinear Programming II: Unconstrained Optimization Techniques

6.2.4 Advantages of Random Search Methods

1.These methods can work even if the objective function is discontinuous and

nondifferentiable at some of the points.

2.The random methods can be used to find the global minimum when the objective

function possesses several relative minima.

3.These methods are applicable when other methods fail due to local difficulties

such as sharply varying functions and shallow regions.

4.Although the random methods are not very efficient by themselves, they can be

used in the early stages of optimization to detect the region where the global

minimum is likely to be found. Once this region is found, some of the more effi-

cient techniques can be used to find the precise location of the global minimum

point.

6.3 Grid Search Method

This method involves setting up a suitable grid in the design space, evaluating the

objective function at all the gird points, and finding the grid point corresponding to

the lowest function value. For example, if the lower and upper bounds on theith

design variable are known to beliandui, respectively, we can divide the range(li, ui)

intopi− equal parts so that 1 x(i^1 ), xi(^2 ),... , xi(pi)denote the grid points along thexi

axis (i= 1 , 2 ,... , n). This leads to a total ofp 1 p 2 · · ·pngrid points in the design

space. A grid withpi= is shown in a two-dimensional design space in Fig. 6.4. The 4

grid points can also be chosen based on methods of experimental design [6.4, 6.5].

It can be seen that the grid method requires prohibitively large number of function

evaluations in most practical problems. For example, for a problem with 10 design

Figure 6.4 Grid withpi=4.