Evolution and Optimum Seeking

Random Strategies 95
the expectation value for the next set of random trials. In contrast to other procedures, the
data from the other trials are not exploited to construct a linear or even quadratic model
from which to calculate a best possible step (e.g., Brooks and Mickey, 1961 Aleksandrov,
Sysoyev, and Shemeneva, 1968 Pugachev, 1970). For small and a large number of
samples, the best value will in any case fall in the locally most favorable direction. In
order to approach a solution with high accuracy, the variance 2 must be successively
reduced. Brooks, however, gives no practical rule for this adjustment. Many algorithms
have since been published that are extensions of Brooks' basic concept of the creeping
random search. Most of them no longer choose the best of several trials they accept each
improvement and reject each worsening (Favreau and Franks, 1958 Munson and Rubin,
1959 Wheeling, 1960).
The iteration rule of a creeping random search is, for the minimum search:
x (k+1) =
( x (k) + z (k) if F (x (k) + z (k) ) F (x (k) ) (success)
x (k) otherwise (failure)
The random vector z (k) ,which in this notation e ects the change in the state vector x,
belongs to an n-dimensional (0 2 ) normal distribution with the expectation value =0
and the variance 2 , which in the simplest case is the same for all components. One can
thus regard , or better p n, as a kind of average step length. The direction of z (k) is uniformly
distributed in IR n , i.e., purely random. Gaussian distributions for the increments
are also used by Bekey et al. (1966), Stewart, Kavanaugh, and Brocker (1967), and De
Graag (1970). Gonzalez (1970) and White (1970) use instead of a normal distribution a
uniform distribution that covers a small region in the form of an n-dimensional cube centered
on the starting point. This clearly favors the diagonal directions, in which the total
step lengths are on average a factor p n greater than in the coordinate directions. Pierre
(1969) therefore restricts the uniformly distributed random probe to an n-dimensional
hypersphere of xed radius. Rastrigin (1960{1972) gives the total step length
s =
vu
u
t nX
a xed value. Instead of the normal distribution he thus obtains a circumferential or
hypersphere-surface distribution. In addition, he repeats the evaluation of the objective
function when there is a failure in order to reduce the e ect of stochastic perturbations.
Taking two model functions
F1(x) = F1(x1:::xn) =
F2(x) = F2(x1:::xn) =
i=1
nX
xi
i=1
vu
u
t nX
i=1
z 2 i
x 2 i
(inclined plane)
(hypercone)
he investigates the average convergence rate of his strategy and compares it with that
of an experimental gradient method, in which the partial derivatives are approximated
by quotients of di erences obtained from exploratory steps . He shows that for a linear