Evolution and Optimum Seeking

92 Random Strategies
rearranging Equation (4.1), the number of trials is obtained
N =
ln (1 ; p)
ln (1 ; v
V )
(4.2)
that is required in order to place with probability p at least one trial in the volume v.
Brooks concludes from this that the cost is independent ofthenumber of variables. In
their criticism Hooke and Jeeves (1958) point out that it is not feasible to consider the
accuracy in terms of the volume ratio for problems with many variables. For n = 100
parameters, a volume ratio of v
=0:1 corresponds to a length ratio of the side length
V
D of V and d of v of
s
d n v
= ' 0:98
D V
This means that the uncertainty in the variables xi is 98% of the original interval [aibi],
although the volume containing the optimum has been reduced to one tenth of the original.
Shimizu (1969) makes the same mistake as Brooks and attempts to implement the strategy
for problems with more general constraints.
A comparison of the pure random search and deterministic search methods known at
the time for experimental optimization problems (Brooks, 1959) also shows no advantage
of the stochastic strategy. The test only covers four di erent objective functions, each
with two variables. Brooks then recommends applying his random method if the number
of parameters is large or if the determination of objective function values is subject to
large perturbations. McArthur (1961) concludes on the basis of numerical experiments
that the random strategy is also preferable for complicated problem structures. Just this
circumstance has led to the use, even today, of the pure random search, often called
the Monte-Carlo method, for example in computer optimization of building construction
(Golinski and Lesniak, 1966 Lesniak, 1970 Hupfer, 1970).
In principle, all the trials of the simple random strategy can be made simultaneously.
It is thus numbered among the simultaneous optimization methods. The decision to
choose a particular state vector of variables does not depend on the results of preceding
trials, since the probability of scoring according to the uniform distribution is the same at
all times. However, in applications on the traditional, serially operating computers, the
trials must be made sequentially. This can be used to advantage by storing the current
best value of the objective function and its associated variable value. In Chapter 3,
Section 3.1.1 and 3.2 the grid or tabulation method was referred to as optimal in the
minimax sense. The blind random strategy should thus not be any better. De ning the
interval length Di = bi ; ai for the variable xi, with required accuracy di, and assuming
that all the Di = D and di = d for i =1(1)n, then for the volume ratio in Equations
(4.1) and (4.2)
v
V =
If v
is small, which when there are many variables must be the case, one can use the
V
approximation
ln (1 + y) ' y for y 1
d
D
! n