An Introduction to Genetic Algorithms - Boente

(4.8)
(This is simply the definition of proportional selection.) Thus, given and F, we can easily find , and
vice versa. Vose and Liepins presented most of their results in terms of .
Given these preliminaries, Vose and Liepins's strategy is to define a single "operator" G such that applying G
to will exactly mimic the expected effects of running the GA on the population at generation t to form the
population at generation t + 1:
(4.9)
Then iterating G on will give an exact description of the expected behavior of the GA. (This is quite
similar to the types of models developed in population genetics by Fisher and others; see, e.g., Ewens 1979.)
To make this clearer, suppose that the GA is operating with selection alone (no crossover or mutation). Let
E(x) denote the expectation of x. Then, si(t) is the probability that i will be selected at each selection step,
Let ~ mean that and differ only by a scalar factor. Then, from equation 4.8, we have
which means
Chapter 4: Theoretical Foundations of Genetic Algorithms
This is the type of relation we want (i.e., of the form in equation 4.9), with G =F for this case of selection
alone.
These results give expectation values only; in any finite population, sampling errors will cause deviation from
the expected values. In the limit of an infinite population, the expectation results are exact.
Vose and Liepins included crossover and mutation in the model by defining G as the composition of the
fitness matrix F and a "recombination operator" that mimics the effects of crossover and mutation. (Vose
and Liepins use the term "recombination" to encompass both the crossover and mutation step. I will adopt this
usage for the remainder of this section.) One way to define is to find ri,j(k),the probability that string k
will be produced by a recombination event between string i and string j, given that i and j are selected to mate.
If ri,j(k) were known, we could compute
In words, this means that the expected proportion of string k in generation t + 1 is the probability that it will be
produced by each given pair of parents, times those parents' probabilities of being selected, summed over all
possible pairs of parents.
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