Evolution and Optimum Seeking

Numerical Comparison of Strategies 231
the measured computation times for all three problems are equal. The results show that
Rechenberg's (1973) theory of the rate of progress, which does not assume a quadratic
objective function but merely concentric hypersphere contour surfaces, is valid over a
wide range of conditions. Even more surprising, however, is the behavior of the (10
, 100) evolution method with recombination in the solution of Problems 3.4 and 3.6,
whose objective functions have discontinuous rst derivatives, i.e., their contour surfaces
display sharp edges and corners. The mixing of the components of variables representing
individuals on di erent sides of a discontinuity appears sometimes to have a kind of
smoothing e ect. In any case it can be seen that the strategy with recombination needs
no more computation time or objective function calls for Problems 3.4 and 3.6 than for
Problems 1.1, 3.1, and 3.2.
With all the methods under test, the computation times for solving Problem 3.7 are
about twice as high as those measured in the simple quadratic case. Only the simplex
method is signi cantly more demanding of time. Since the search simplex frequently
collapses in on itself it must repeatedly be reinitialized.
Since Problem 3.3 could only be tackled with 3, 10, and 30 variables it is not easy to
analyze the resulting data. In addition, the dependence of the increase in di cultyonthe
number of parameters is not so clear-cut in this problem. Nevertheless the results seem to
indicate that at least the number of objective function calls, in many strategies, increases
with n in a way similar to that in the pure quadratic Problem 1.2. Because an objective
function evaluation takes about O(n 2 ) operations in Problem 3.3, the total cost generally
increases as one higher power of n than in Problem 1.2. The cost of the variable metric
strategy and both versions of the (10 , 100) evolution strategy seems to increase even more
rapidly. In the latter case there is a suspicion that the chosen initial step lengths are too
large for this problem when there are very many variables. Their reduction to a suitable
size then takes a few additional generations. The twomembered evolution strategy, which
is able to adjust unsuitable initial step lengths relatively quickly, needed about the same
number of mutations for both Problems 1.2 and 3.3. Since only one experiment per
strategy and number of variables was performed, the e ect of the particular sequence
of random numbers on the recorded computation times is not known. The particularly
advantageous behavior of the DFPS method on exactly quadratic objective functions is
clearly wasted once the problem deviates from this model structure in fact it seems that
the search process is appreciably held back byaninterpretation of the measured data in
terms of an inappropriate internal model.
So far we have only discussed the results for the seven unconstrained problems, since
they were amenable to solution by all the search strategies. Problem 3.8, with constraints,
corresponds to the second model function (corridor model) for which Rechenberg (1973)
has obtained theoretically the rate of progress of the two membered evolution strategy
with optimal adaptation of variances. According to his analysis, one expects a linear rate
of convergence increasing with the width of the corridor and inversely proportional to
the number of variables. The results of the third set of tests con rm that the number
of mutations or generations increases linearly with n if the width of the corridor and
the reference distance to be covered are held constant. The picture for the Rosenbrock
strategy is as usual: the time consumption increases as O(n 3 ) again. The point atn =75