Evolution and Optimum Seeking

96 Random Strategies
problem structure like F1 the random strategy needs only O( p n) trials, whereas the
gradient strategy needs O(n) trials to cover a prescribed distance. For n>3, the random
strategy is always superior to the deterministic method. Whereas Rastrigin shows that the
random search always does better than the gradient searchin the spherically symmetric
eld F2, Movshovich (1966) maintains the opposite. The discrepancy can be traced to
di ering assumptions about the choice of step length (see also Yvon, 1972 Gaviano and
Fagiuoli, 1972).
To choose suitable step lengths or variances poses the same problems for sequential
random searches as are familiar from deterministic strategies. Here too, a closely related
problem is to achieve global convergence with reference to a suitable termination rule, the
convergence criterion, and with a degree of reliability. Khovanov (1967) has conceived
an individual manner of controlling the random step lengths. He accepts every random
change, irrespective of success or failure, increases the variance at each failure and reduces
it otherwise. The objective is to increase the probability of lingering in the more promising
regions and to abandon states that are irrelevant to the optimum search. No applications
of the strategy are known to the author. Favreau and Franks (1958), Bekey et al. (1966),
and Adams and Lew (1966) use a constant ratio between i and xi for i =1(1)n. This
measure does have the e ect of continuously altering the \step lengths," but its merit is
not obvious. Just because a variable value xi is small in no way indicates that it is near to
the extreme position being sought. Karnopp (1961) was the rst to propose a step length
rule based on the number of successes or failures, according to which the i or s are all
uniformly reduced or enlarged such that a success always occurs after two or three trials.
Schumer (1967), and Schumer and Steiglitz (1968), submit Rastrigin's circumferential
random direction method to a thorough examination by probability theory. For the
model
F3(x) =
nX
i=1
x 2
i = r 2
with the condition n 1 and the continuously optimal step length
s ' 1:225 r
p n
they obtain a rate of progress ', which istheaverage distance covered in the direction of
the objective (minimum) per random step:
' ' 0:203 r
n
and a success rate ws which istheaverage number of successes per trial:
ws ' 0:270
They are only able to treat the general quadratic case theoretically for n = 2. Their
result can be interpreted in the sense that ' is dependent on the smallest radius of
curvature of the elliptic contour passing through r. Since neither r nor s can be assumed
to be known in advance, it is not clear how tokeep to the optimal step length. Schumer
and Steiglitz (1968) give an adaptive method with which the correct size of s can be