Evolution and Optimum Seeking

Genetic Algorithms 157
out any change) between the four di erent genetic states 00 01 10 11 (e.g., phenotypes
0 1 2 3), those from 01 to 10 and from 10 to 01 are the most improbable ones despite their
phenotypic vicinity. Let pm = 10 ;3 then q 2 = 0:998001pq = 0:000999 and p 2 =
0:000001:
5.3.4 Reproduction and Selection
Whether selection is the rst or last operator in the generation loop of EAs should not
matter except for the rst iteration. The di erence in this respect between ESs and GAs,
however, is that both mingle several aspects of the generation transition. Let us look rst,
therefore, at the biological facts to be modelled by a selection operator.
An o spring may ormay not be able to survive the time span between birth and
reproduction. If it is vital up to its reproductive ageitmayhave varying numbers of
o spring with one or more partners of its own generation. Thus, the term \selection" in
EAs comprises at least three di erent aspects:
Survival to adult state (ontogeny)
Mating behavior (perhaps including promiscuity)
Reproductive activity
Both ESs and GAs select parents for each o spring anew, thus modelling maximal
promiscuity. GAs assign higher mating and reproductive activities to individuals with
better objective function values (both for proportional as well as linear or other ranking
selection). But even the worst o spring of generation g may become parents for generation
g +1. The probability, however, may bevery low. If this is the case, most o spring
are descendants of a few best parents only. The corresponding loss of diversity inthe
population may lead to premature stagnation (not convergence!) of the evolutionary
seeking process. Reducing the proportionality factor in the selection function, on the
other hand, ultimately leads to random walk behavior. This enhances the reliability in
multimodal situations, but reduces the convergence velocity and the precision of locating
the optimum.
For proportional selection, after Holland derived from an analogy to the game-theoretic
multiarmed bandit problem, the average number of o spring for an individual with genotype
ak, phenotype xk, and vitality f(xk) is
(f(xk))
(ak) = ps(ak) =
=
1 X
(f(xi))
k
The transformation (f) is necessary for introducing the proportionality factor mentioned
aboveaswell as for dealing with negativevalues of the objective function. ps often is called
the survival probability, which is misleading. No parent really survives its generation
except in an elitist GA version. Then the best parent isputinto the next generation
i=1