Evolution and Optimum Seeking

148 Evolution Strategies for Numerical Optimization
(e.g., by a random value very far from the expectation value) so much reduced in size that
the associated variable xi can now hardly be changed. The total change in the vector x is
then roughly speaking within an (n ; 1)-dimensional subspace of IR n .Contrary to what
one might hope, that such a descendant would have less chance of surviving than others,
it turns out that the survival of such a descendant is actually favored. The reason is that
the rate of progress with an optimal step length is proportional to 1=n. If the number
of variables n decreases, the rate of convergence, together with the optimal step length,
increases. The optimum search therefore only proceeds in a subspace of IR n . Not until
the only improvement in the objective function entails changing the variable that has
hitherto been omitted from the variation will the mutation-selection mechanism operate
to increase its associated variance and so restore it to the range for which noticeable
changes are possible.
The minimum search proceeds by jumps in the value of the objective function and
with rates of progress that vary alternately above and below whatwould otherwise be
smooth convergence. Such unstable behavior is most pronounced when , the number
of parents, is small. With su ciently large the reserve of step length combinations
in the gene pool is always big enough to avoid overadaptation, or to compensate for it
quickly. From an experimental study (Schwefel, 1987) the conclusion could be drawn
that punctuated equilibrium evolution (Gould and Eldredge, 1977, 1993) can be avoided
by using a su ciently large population ( >1) and a su ciently low selection pressure
( = ' 7). A further improvement can be made by using as the starting point inthe
variation of the step lengths the currentaverage of two parents' variances, rather than the
value from only one or the other parent. This measure too has its biological justi cation
it represents an imitation of what is called intermediary recombination (instead of discrete
recombination).
In this context chromosome mutations should be very e ective, those in which for
example, the positions of two individual step lengths are exchanged. As well as the haploid
scheme of inheritance on which the presentwork is based, some forms of life also exhibit the
diploid scheme. In this case each individual stores two sets of variable values. Whilst the
formation of the phenotype only makes use of one allele, the production of o spring brings
both alleles into the gene pool. If both alleles are the same one speaks of homozygosity,
otherwise of heterozygosity. Heterozygote alleles enlarge the set of variants in the gene
pool and thus the range of possible combinations. With regard to the stability of the
evolutionary process this also appears to be advantageous. The true gain made by diploidy
only becomes apparent, however, when the additional evolutionary factors of recessiveness
and dominance are included. For multiple criteria optimization, the usefulness of this
concept has been demonstrated by Kursawe (1991, 1992). Many possible extensions of
the multimembered scheme have yet to be put into practice. To nd their theoretical
e ect on the rate of progress, one would rst have to construct a theory of the ( , )
strategy for > 1. If one goes beyond the = 1 scheme followed here, signi cant
di erences between approximate theory and simulation results arise for >1 because of
the greater asymmetry of the probability distribution w(s 0 ).