Evolution and Optimum Seeking

156 Evolution Strategies for Numerical Optimization
Table 5.1: Two point crossover within a GA and its e ect on the
phenotypes
Bit strings Phenotype
Parent 1 0111 1100 7 12
Parent 2 1000 1011 8 11
Two point crossover
O spring 1 0000 1000 0 8
O spring 2 1111 1111 15 15
completely contradicts the proverbial saying that the apple never falls far from the tree.
Table 5.1 shows a simple situation with two parents producing two o spring by means of
two point crossover, on a bit string of length 8, and encoding two phenotypic variables
in the range [0 15]in the standard Boolean form. Neither discrete nor intermediary
recombination within ESs can be that disruptive intermediary recombination always
delivers phenotypic values for the o spring between those of their parents. The assumption
that mutations are not necessary for the GA process may even stem from that disruptive
character of recombination that permits crossover points not only at the boundaries of
meaningful parental information but also within the genes themselves.
ESs obey the general rule, that mutations are undirected, by means of using normally
distributed changes with zero mean{even in the case of correlated mutations. That this
is not so for GAs can easily be seen from Figure 5.16. Without selection, the GA process
thus provides biased genetic drift, depending on the actual situation.
Table 5.2 presents the probability transition matrix for one phenotypic integer variable
xi in the range [0 3]encoded by means of two bitsonly.Let
p = pm
1
2
single bit inversion probability and
q =1; pm probability of not inverting the bit
From Table 5.2 it is obvious that among all possible transitions (except for those with-
Table 5.2: Transition probabilities for mutations within a GA
xi new
Genotype 00 01 10 11
Phenotype 0 1 2 3
00 0 q 2 pq pq p 2
xi old 01 1 pq q 2 p 2 pq
10 2 pq p 2 q 2 pq
11 3 p 2 pq pq q 2