Evolution and Optimum Seeking

Numerical Comparison of Strategies 211
is in uenced by random numbers, the same rule was followed: namely, out of three tests
the one with the best end result was accepted. In contrast to the strategy of Box, however,
the evolution methods prove to be less dependent on the actual sequence of random
numbers. This is especially true of the multimembered versions. Recombination almost
always improves the chance of getting very close to the desired solutions. Fatal errors
due to exceeding the maximum number range or dividing by zero do not occur by virtue
of the simple computational operations in these strategies. Discontinuities in the partial
derivatives, saddle points, and the like have noobvious adverse e ects. The search does,
however, become rather time consuming when the minimum is reached via a long, narrow
valley. The step lengths or variances that are set in this case are very small and impose
slow convergence in comparison to methods that can perform a line search along the
valley. The average rate of progress of an evolution strategy is not, however, a ected by
bends in the valley, which would retard a one dimensional minimization procedure. Line
searches only a ord a signi cant advantage to the rate of progress if there are directions in
the space along which successful steps can be made of a size that is large compared to the
local radius of curvature of the objective function contour surface. Examples are provided
by Problems 2.14, 2.15, and 2.28. In these cases, long before reaching the minimum the
optimal variances of the evolution methods have reached the lower limit as determined
by the machine accuracy. The desired solution cannot therefore be approximated to the
required accuracy. In Problems 2.14 and 2.15 the computation time limit did not allow
the convergence criterion to be satis ed although it was actually progressing slowly but
surely, the search was terminated.
Di culties with the termination rule based on function values only occurred in the
solution of one type of problem (Problems 2.11, 2.12) using the (10 , 100) evolution strategy
with recombination. The multimembered method selects the 10 best individuals of a
generation only from the current 100 descendants. Their 10 parents are not included in
the selection process, for reasons associated with the step length adaptation. In general,
the objective function value of the best descendant is closer to the solution than that
of the best parent. In the case of the two problems referred to above, this is initially
the case. As the solution is approached, however, it happens more and more frequently
that the best value occurring in a generation is lost again. This is related to the fact
that because of rounding errors in evaluating values near the minimum, the objective
function behaves practically stochastically. Thus the population wanders around in the
neighborhood of the (quasi-singular) optimal solution without being able to satisfy the
convergence criterion. These di culties do not beset the other search methods, including
the multimembered evolution without recombination, because they do not come nearly so
close to the optimum. The fact that the third problem of the same type (Problem 2.10)
is solved without di culties in a nite time, even with recombination, can be considered
a uke. Here too the minimum was reached long before the termination criterion was
satis ed. On the whole, the multimembered evolution strategy with recombination is the
surest and safest of all the search methods tested. In only 5 out of 28 cases is the solution
not located exactly, and the greatest deviations of the variables were in the accuracy class
=10 ;4 .