Genetic Algorithms

<script type="text/javascript" src="path-to-MathJax/MathJax.js"></script>

Bold textGenetic Algorithms

1. Genetic algorithms (GAs): basic form

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

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

where H is a subset of the Euclidean space Italic textR.

The general problem is to find

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

where Italic textX

is a vector of the decision variables and Italic text f is the objective function.

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

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

The mapping Italic text c is not necessarily surjective. The range of Italic text c determine the subset of Italic text^{Superscript text} Al available for exploration by a GA. The range of Italic text c, Italic text Ξ

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

is needed to account for the fact that some strings in the image Italic text^{Superscript text} Al under Italic text c may represent invalid solutions to the original problem.

The string length Italic text l depends on the dimensions of both Italic text H and Italic text^{Superscript text} Al, with the elements of the string corresponding to genes and the values to 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 Italic text c is a bijection. The important property of bijections as they apply to GAs is that bijections have an inverse, i.e., there is a unique vector Italic text x for every string and a unique string for each Italic text x.

2. Genetic algorithms and their Operators

Let H be a nonempty set (the individual or search 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 \[\left\{{{r}^{(i)}} \right\}\]

Define the collection Italic textμ (the number of individuals) via _{Subscript text}Italic textHμ. The population transforms are denoted by

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

where \[\mu \in \mathbb{N}\ of recombination operators τ(i) : \[X_{r}^{(i)}\ to T(\Omega _{r}^{(i)},T\left( {{H}^{{{u}^{(i)}}}},{{H}^{u{{'}^{(i)}}}} \right))\], 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), and \[\Theta _{s}^{(i)}\ in X_{s}^{(i)}\] (the selection parameters).

Some GA methods generate populations whose size not equal to their predecessors’. The following expression

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

can accommodate populations that contain the same or different individuals. This mapping has the ability to represent all population sizes, genetic operators, and parameters as sequences.

The execution of a GA typically begins by randomly sampling with replacement from ^{Superscript text}Italic text Al. The resulting collection is the initial population, denoted by Italic text P(0). In general, a population is a collection \[P=({{a}_{1}},{{a}_{2}},...,{{a}_{\mu }})\]of individuals, where\[{{a}_{i}}\in {{A}^{l}}\], and populations are treated as n-tuples of individuals. The number of individuals (μ) is defined as the population size.

Using the work of Lamont and Merkle (Lamont, 1997) we describe the termination criteria and the other genetic operators (GOs).

Since Italic text H is a nonempty set,\[c:{{A}^{l}}\to H\], and\[f:H\to \mathbb{R}\], the fitness scaling function can be defined as \[{{T}_{s}}:\mathbb{R}\to \mathbb{R}\]and a related fitness function as\[\Phi \triangleq {{T}_{s}}\circ f\circ c\]. In this definition it is understood that the objective function Italic textf is determined by the application, while the specification of the decoding function Italic textc[1] and the fitness scaling function ^{Subscript text}Italic textTs are design issues.

Following initialization, execution proceeds iteratively. Each iteration consists of an application of one or more GOs. The combined effect of the GOs applied in a particular generation $t\in N$ is to transform the current population Italic text P(t) into a new population Italic text P(t+1).

In the population transformation $\mu ,{\mu}'\ in {{\mathbb{Z}}^{+}}$(the parent and offspring population sizes, respectively). A mapping $T:{{H}^{\mu }}\ to {{H}^{{{\mu }'}}}$ is called a population transformation (Italic text PT). If $T(P)={P}'$, then Italic textP is a parent population and ^{Superscript text}Italic textP/ is the offspring population. If $\mu ={\mu }'$,then it is called simply the population size.

The Italic text PT resulting from an GO often depends on the outcome of a random experiment. This result is referred to as a random population transformation (Italic textRPT or random PT). To define Italic textRPT, let $\mu \in {{\mathbb{Z}}^{+}}$and $\Omega $ be a set (the sample space). A random function $R:\Omega \to T({{H}^{\mu }},\bigcup\limits_{{\mu }'\ in {{\mathbb{Z}}^{+}}}^{{}}{{{H}^{{{\mu }'}}}})$ is called an Italic textRPT. The distribution of Italic textPTs resulting from the application of a GO depends on the operator parameters; in other words, a GO maps its parameters to an Italic textRPT.

Now that both the fitness function and Italic text RPT have been defined, the GO can be defined in general once again, this time in more detail: let$\mu \in {{\mathbb{Z}}^{+}}$, Italic textX be a set (the parameter space) and $\Omega $ a set. The mapping $\Zeta :X\to T\left( \Omega ,T\left[ {{H}^{\mu }},\bigcup\limits_{{\mu }'\in {{\mathbb{Z}}^{+}}}^{{}}{{{H}^{{{\mu}'}}}} \right] \right)$ is a GO. The set of GOs is denoted as $GAOP\left( H,\mu ,X,\Omega \right)$.

There are three common GOs: recombination,mutation, and selection. These three operators are roughly analogous to their similarly named counterparts in genetics. The application of them in GAs is strictly Darwin-like in nature, i.e., “survival of the fittest.”

For the recombination operator let $r\in GAOP\left( H,\mu ,X,\Omega \right)$. If there exists $P\in {{H}^{\mu }},\Theta \in X$, and $\omega \in \Omega $, such that one individual in the offspring population ${{r}_{\Theta }}\left( P \right)$ depends on more than one individual of P,then r is referred to as a recombination operator.

For the mutation operator let $m\in GAOP\left( H,\mu ,X,\Omega\right)$. If for every $P\in {{H}^{\mu }}$, for every $\Theta \in X$, for every $\omega\in \Omega $, and if each individual in the offspring population ${{m}_{\Theta}}\left( P \right)$ depends on at most one individual of P, then m is called a mutation operator.

Finally, for the selection operator let $s\in EVOP\left( H,\mu ,X\times T\left(H,\mathbb{R}),\Omega \right) \right)$. If $P\in {{H}^{\mu }}$,$\Theta \in X$,$\Phi :H\to\mathbb{R}$in all cases, and satisfies $a\in {{s}_{\left( \Theta ,\Phi \right)}}(P)\Rightarrow a\in P$, then s is a selection operator.

Bold textEndnotes

[1] If the domain of c is total, i.e., the domain of c is all of A I, c is called a decoding function. The mapping of c is not necessarily surjective. The range of c determines the subset of Al available for exploration by the evolutionary algorithm.