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=Genetic Algorithms=
 
  
==Genetic algorithms (GAs): basic form==
 
 
A generic GA (also known as an evolutionary algorithm [EA]) assumes a discrete search space $H$ and a function
 
\[f:H\to\R,\]
 
where $H$ is a subset of the Euclidean space $\R$.
 
 
The general problem is to find
 
\[\arg\underset{X\in H}{\mathop{\min }}\,f\]
 
where $X$ is a vector 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 variables themselves. The  vector $X$ is represented by a string (or chromosome) $s$ of length $l$ made up 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 $A^l$ available for exploration by a GA.  The range of $c$, $\Xi$
 
\[\Xi\subseteq {{A}^{l}}\]
 
is  needed to account for the fact that some strings in the image $A^l$ under $c$  may represent invalid solutions to the original problem.
 
 
The  string length $l$ depends on the dimensions of both $H$ and $A^l$, 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 $c$ 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 $x$ for every string and a unique string for each $x$.
 
 
==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,
 
\[\iota :\cup_{i=1}^{\infty}{{({{H}^{u}})}^{(i)}}\to\{\text{true},\text{false}\}\]
 
(the termination  criteria),
 
\[\chi\in\{\text{true},\text{false}\},\]
 
$r$ a sequence
 
\[\left\{{{r}^{(i)}}  \right\}\]
 
 
Define the collection $\mu$ (the  number of individuals) via $H_\mu$. The population transforms are denoted by
 
\[T:{{H}^{\mu }}\to {{H}^{\mu }}\]
 
where
 
\[\mu \in \mathbb{N}\]
 
of recombination operators $\tau(i)$:
 
\[X_{r}^{(i)}\to T(\Omega_{r}^{(i)},T\left( {{H}^{{{u}^{(i)}}}},{{H}^{u{{'}^{(i)}}}} \right)),\]
 
$m$ 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):
 
\[X_{s}^{(i)}\times T(H,\mathbb{R})\to T(\Omega_{s}^{(i)},T(({{H}^{u{{'}^{(i)+\chi {{\mu}^{(i)}}}}}}),{{H}^{{{\mu  }^{(i+1)}}}})),\]
 
\[\Theta_{r}^{(i)}\in X_{r}^{(i)}\]
 
(the  recombination parameters),
 
\[\Theta_{m}^{(i)}\in X_{m}^{(i)}\]
 
(the  mutation parameters), and
 
\[\Theta_{s}^{(i)}\in X_{s}^{(i)}\]
 
(the  selection parameters).
 
 
To account for the cases where populations whose size not equal to their predecessors’, the following expression
 
\[T:{{H}^{\mu}}\to {{H}^{{{\mu }'}}}\]
 
accommodates  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 random sampling with replacement  from $A^l$.  The  resulting collection is the initial population, denoted by $P(0)$. 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 ($\mu$) is defined as the population size.
 
 
Following upon the  work of Lamont and Merkle (Lamont, 1997) and others, the termination  criteria and the other genetic operators(GOs)will be defined in more  detail.
 
 
Since $H$ 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 $f$ is determined by the application, while the specification of  the decoding function $c[1]$ and the fitness scaling  function $T_s$ are  design issues.
 
 
 
After initialization, the  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 a transformation of 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 text''P is the parent population and  <sup>Superscript text</sup>''Italic text''P/ the offspring  population. If $\mu ={\mu }'$,then it is called the population size.
 
 
The  ''Italic text''PT resulting from a GO often depends on the outcome of a  random experiment. This result is referred to as a random population  transformation (''Italic text''RPT or random PT). To define ''Italic  text''RPT, 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 a ''Italic text''RPT. The distribution of ''Italic  text''PTs resulting from the application of a GO depends on the operator  parameters; in other words, a GO maps its parameters to a ''Italic  text''RPT.
 
 
Now that both the fitness function and  ''Italic text''RPT have been defined, let$\mu \in {{\mathbb{Z}}^{+}}$,  ''Italic text''X 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 text'''Endnotes
 
 
[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.
 

Revision as of 20:32, 18 September 2012

How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=27707