Evolutionary Computation : A Unified Approach

6.4. REPRODUCTION-ONLY MODELS 143
6.4.1.1 Reproduction for Fixed-Length Discrete Linear Genomes
Recall that all of the models developed so far in this chapter have assumed a finite set of
r genotypes. A standard additional assumption is that the genomes are linear structures
consisting of a fixed number of L genes, each of which can take on only a finite set of values
(alleles). This assumption does not significantly widen the gap between theory and practice,
since many EAs use fixed-length linear representations. Making this assumption allows for
an additional analytical perspective, in which one can focus on a particular gene position
(locus) j and study the distribution of its k j allele values a j1 ...a jkj in the population. Of
particular interest is how the frequency f(a ji ) of allele a ji in the population changes over
time, which provides additional insight into the effects of reproductive variation.
Traditional Two-Parent Crossover
The traditional two-parent crossover operator produces genetic variation in the offspring by
choosing for each locus i in the offspring’s genome whether the allele value at locus i will
be inherited from the first parent or from the second one. We can characterize the result of
this process as a binary string of length L, in which a value of 0 at position i indicates that
the offspring inherited allele i from the first parent, and a value of 1 at position i indicates
that the allele value was inherited from the second parent. If we imagine this assignment
process as proceeding across the parents’ genomes from left to right, then the string 0000111
represents the case in which this reproductive process assigned to the offspring the first 4
alleles from parent one, and then “crossed over” and assigned the remaining three alleles
from parent two. Patterns of this sort, in which offspring inherit an initial allele segment
from one parent and a final allele segment from the other parent, are generated by the
familiar 1-point crossover operator used in canonical GAs. Similarly, a 2-point crossover
operator generates patterns of the form 00011100, and an (L-1)-point crossover operator
simply alternates allele assignments. An interesting alternative to n-point crossover is the
“uniform” crossover operator (Syswerda, 1989), in which alleles are inherited by flipping an
unbiased coin at each locus to determine which parent’s allele gets inherited, thus uniformly
generating all possible crossover patterns.
Recall from the analysis in section 6.3 that selection-only EAs using uniform (neutral)
selection converge to a fixed point. With infinite-population models, that fixed point is
the initial population P (0). So, what happens if we add a two-parent crossover operator
to these uniform selection models Clearly, every two-parent crossover operator has the
ability to introduce new genomes into the population that were not present in the initial
population P (0). However, since the offspring produced can only consist of genetic material
inherited from their parents on a gene-by-gene basis, no new gene values (alleles) can be
introduced into the population. Hence, any new genomes must consist of recombinations
of the alleles present in P (0), and all population trajectories are restricted to points in the
simplex representing those populations whose alleles are present in P (0). Thus, infinite
population EAs with neutral selection and two-parent crossover no longer have P (0) as
their fixed point. Rather, they converge to a rather famous fixed point called the “Robbins
equilibrium” (Geiringer, 1944). To express this more precisely, let L be the length of a
fixed-length genotype, and let a ij represent the allele value of the j th gene of genotype i.
Then, the population fixed point is given by: