Evolution and Optimum Seeking

94 Random Strategies
knowledge of the approximate position of the optimum, the subspaces will be assigned
di erent sizes (Idelsohn, 1964). The original uniform distribution is thereby replaced by
one with a greater density in the neighborhood of the expected optimum. Karnopp (1961,
1963, 1966) has treated this problem in detail without, however, giving any practical
procedure. Mathematically based investigations of the same topic are due to Motskus
(1965), Hupfer (1970), Pluznikov, Andreyev, and Klimenko (1971), Yudin (1965, 1966,
1972), Vaysbord (1967, 1968, 1969), Taran (1968a,b), Karumidze (1969), and Meerkov
(1972). If after several (simultaneous) samples the search is continued in an especially
promising looking subregion, the procedure becomes sequential in character. Suggestions
of this kind have been made for example by McArthur (1961), Motskus (1965), and
Hupfer (1970) (shrinkage random search). Zakharov (1969, 1970) applies the stochastic
approximation for the successive shrinkage of the region in which Monte-Carlo samples
are placed. The most thoroughly worked out strategy is that of McMurtry and Fu (1966,
probabilistic automaton see also McMurtry, 1965). The problem considered is to adjust
the variable parameters of a control system for a dynamic process in such away that the
optimum of the system is found and maintained despite perturbations and (slow) drift
(Hill, McMurtry, and Fu, 1964 Hill and Fu, 1965). Initially the probabilities are equal
for all subregions, at the center of which the function values are measured (assumed to be
stochastically perturbed). In the course of the iterations the probability matrix is altered
so that regions with better objective function values are tested more often than others.
The search ends when only one subregion remains: the one with the highest probability
of containing the global optimum. McMurtry and Fu use a so-called linear intensi cation
to adjust the probability matrix. Suggestions for further improving the convergence rate
have been made by Nikolic and Fu (1966), Fu and Nikolic (1966), Shapiro and Narendra
(1969), Asai and Kitajima (1972), Viswanathan and Narendra (1972), and Witten (1972).
Strongin (1970, 1971) treats the same problem from the point of view of decision theory.
All these methods lay great emphasis on the reliability of global convergence. The
quality of the approximation depends to a large extent on the number of subdivisions
of the n-dimensional region under investigation. High accuracy requirements cannot be
met for many variables since, at least initially, the number of subregions to investigate
rises exponentially with the number of parameters. To improve the local convergence
properties, there are suggestions for replacing the midpoint tests in a subvolume by the
result of an extreme value search. This could be done with one of the familiar search
strategies such as a gradient method (Hill, 1969) or any other purely sequential random
search method (Jarvis 1968, 1970) with a high convergence rate, even if it were only
guaranteed to converge locally. Application, however, is limited to problems with at most
seven or eight variables, as reported.
Another possibility for giving a sequential character to random methods consists of
gradually shifting the expectation value of a random variable with a restricted probability
density distribution. Brooks (1958) calls his proposal of this type the creeping random
search. Suitable random numbers are provided for example by a Gaussian distribution
with expectation value and standard deviation . Starting from a chosen initial condition
x (0) , several simultaneous trials are made, which most likely fall in the neighborhood of the
starting point ( = x (0) ). The coordinates of the point with the best function value form