Genetic Algorithms

Bold textGenetic Algorithms

1. Genetic algorithms (GAs): basic form

A generic GA (also know as an evolutionary algorithm [EA]) assumes a discrete search space H and a function

                                              \[f:H\to\mathbb{R}\],                         

where H is a subsetof the Euclidean space\[\mathbb{R}\]. The general problem is to find

                                              \[\arg\underset{X\in H}{\mathop{\min }}\,f\]

where X is avector of the decision variables and f is the objective function.

With GAs it is customary to distinguish genotype–the encoded representation of the variables–from phenotype–the set of variablesthemselves. The vector X is represented by a string (or chromosome) s of length l madeup of symbols drawn from an alphabet A using the mapping

                                              \[c:{{A}^{l}}\to H\]

The mapping c is not necessarily surjective. The range of c determine the subset of Al available for exploration by a GA. The range of c, Ξ

                                              \[\Xi\subseteq {{A}^{l}}\]

is needed to account for the fact that some strings in the image Al under c may represent invalid solutions to the original problem.

The string length l depends on the dimensions of both H and A, with the elements of the string corresponding to genes and the valuesto alleles. This statement of genes and alleles is often referred to as genotype-phenotype mapping.

Given the statements above, the optimization becomes:

                                     \[\arg\underset{S\in L}{\mathop{\min g}}\,\],

given the function

                                      \[g(s)=f(c(s))\].

Finally, with GAs it is helpful if c is a bijection. The important property of bijections as they applyto GAs is that bijections have an inverse, i.e., there is a unique vector x for every string and a unique stringfor each x.

2. Genetic algorithms and their operators

The following statements about the operators of GAs are adapted from Coello et al.(2002).

· Let H be a nonempty set (the individual orsearch space) · \[{{\left\{{{u}^{i}} \right\}}_{i\in \mathbb{N}}}\] a sequence in \[{{\mathbb{Z}}^{+}}\] (the parent populations) · \[{{\left\{ {{u}^{'(i)}} \right\}}_{i\in\mathbb{N}}}\] a sequence in \[{{\mathbb{Z}}^{+}}\](the offspring population sizes) · \[\phi :H\to \mathbb{R}\] a fitness function: \[\iota :\cup _{i=1}^{\infty}{{({{H}^{u}})}^{(i)}}\to \] {true, false} (the termination criteria) \[\chi\in \]{true, false} · r a sequence \[\left\{ {{r}^{(i)}} \right\}\] of recombination operators τ(i) :\[X_{r}^{(i)}\to T(\Omega _{r}^{(i)} · m a sequence of {m(i)} of mutation operators in mi, \[X_{m}^{(i)}\ to T(\Omega _{m}^{(i)},T\left( {{H}^{{{u}^{(i)}}}},{{H}^{u{{'}^{(i)}}}} \right))\] · s a sequence of {si} selection operators s(i): \[X_{s}^{(i)}\timesT(H,\mathbb{R})\to T(\Omega _{s}^{(i)},T(({{H}^{u{{'}^{(i)+\chi {{\mu}^{(i)}}}}}}),{{H}^{{{\mu }^{(i+1)}}}}))\] · \[\Theta _{r}^{(i)}\in X_{r}^{(i)}\] (the recombination parameters) · \[\Theta _{m}^{(i)}\in X_{m}^{(i)}\] (the mutation parameters) · \[\Theta _{s}^{(i)}\in X_{s}^{(i)}\] (the selection parameters)

Coello et al. define the collection μ (thenumber of individuals) via Hμ.The population transforms are denoted by

                                  \[T:{{H}^{\mu }}\to {{H}^{\mu }}\]

where

                                  \[\mu \in \mathbb{N}\]

However, some GA methods generate populationswhose size is not equal to their predecessors’. In a more general framework

                                  \[T:{{H}^{\mu}}\to {{H}^{{{\mu }'}}}\]

can accommodate populations that contain the same ordifferent individuals. This mapping has the ability to represent all populationsizes, evolutionary operators, and parameters as sequences.