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'''Bold text'''Genetic Algorithms
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{{MSC|90C99,60H25,68Q87}}
  
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Genetic algorithms (GAs) (also known as evolutionary algorithms [EAs]) assume a discrete search space $H$ and a function $f:H\to \R$.  The general problem is to find $\argmin{x\in H}f$, 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 $c:A^l\to H$.  In practice we may need a search space $\Xi\subseteq A^l$ to reflect the fact that some strings in the image  $A^l$ under $c$ may represent invalid solutions to the 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 values to alleles. This statement of genes and alleles is often referred to as genotype-phenotype mapping.  Given the statements above, the optimization becomes one of finding $\argmin{S\in L}$, where the function  $g(s)=f(c(s))$.  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 $x$.
  
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The execution of an GA typically begins by randomly 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 initialization, execution proceeds iteratively.  Each iteration consists of an application of one or more of the genetic operators (GOs): recombination, mutation and selection. The combined effect of the GOs applied in a particular generation $t\in N$ is to transform the current population $P(t)$ into a new population $P(t+1)$.  In the population transformation $\mu$, $\mu'\in\Z^+$ (the parent and offspring population sizes, respectively). A mapping $T:H^\mu\to H^{\mu'}$ is called a PT.  If $T(P)=P'$ then $P$ is a parent population and $P'$ is the offspring population.  If $\mu=\mu'$ it is simply the population size.  The PT resulting from a GO often depends on the outcome of a random experiment. In Lamont and Merkel this result is referred to as a random population transformation (RPT or random PT). To define RPT, let $\mu \in \Z^+$and $\Omega $ be a set (the sample space).  A random function $R:\Omega \to T(H^\mu,\bigcup\limits_{\mu'\in\Z^+} H^{\mu'})$ is called a RPT. The distribution of PTs resulting from the application of a GO depends on the operator parameters.  In other words, a GO maps its parameters to an RPT.  Since H is a nonempty set where $c:A^l\to H$, and $f:H\to\R$, the fitness scaling function can be defined as $T_s:\R\to\R$ and a related fitness function $\Phi \triangleq T_s\circ f\circ c$. In this definition it is understood that the objective function $f$ is determined by the application.
  
1.       Genetic algorithms (GAs): basic form
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Now that both the fitness function and RPT have been defined, the GOs can be defined in general: let $\mu \in \Z^+$, $\aleph$ be a set (the parameter space) and $\Omega $ a set (the sample space). The mapping $X:\aleph \to T\left( \Omega ,T\left[ H^\mu,\bigcup\limits_{\mu'\in \Z^+} H^{\mu'} \right] \right)$ is a GO.  The set of GOs is denoted as $GOP( H,\mu ,\aleph ,\Omega)$.  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.”  In defining the GOs, we will start with recombination.  Let $r\in GOP( H,\mu ,\aleph ,\Omega)$. If there exist $P\in H^\mu$, $\Theta \in \aleph $ and $\omega\in \Omega $ such that one individual in the offspring population $r_\Theta (P)$ depends on more than one individual of $P$ then $r$ is referred to as a recombination operator.  A mutation is defined in the following manner: let $m\in GOP( H,\mu ,\aleph ,\Omega)$. 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 $m_\Theta(P)$ depends on at most one individual of $P$ then $m$ is called a mutation operator.  Finally, for a selection operator let $s\in GOP( H,\mu ,\aleph \times T(H,\R),\Omega ) )$.  If $P\in H^\mu$, $\Theta \in \aleph $, $f:H\to \R $ in all cases, and $s$ satisfies $a\in s_{( \Theta,\Phi)}(P) \Rightarrow a\in P$, then $s$ is a selection operator.
  
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.
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'''References'''
  
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
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[1] Back, T., Evolutionary Algorithms in Theory and Practice, Oxford University Press, 1996
  
                                                    \[c:{{A}^{l}}\to H\]
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[2] Coello Coello, C.A., Van Veldhuizen, D.A. and Lamont, G.B., Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, 2002
  
The mapping c is not necessarily surjective. The range of c determine the subset of Al  available for exploration by a GA.
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[3] Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Professional, 1989
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.
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[4] Holland, J.H., Adaptation in Natural and Artificial Systems, A Bradford Book, 1992
  
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.
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[5] Lamont, L. and Merkle, J.D., "A Random Based Framework for Evolutionary Algorithms", in: T. Back, Proceedings of the Seventh International Conference on Genetic Algorithms, 1997.
  
Given the statements above, the optimization becomes:
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[6] Surry, Patrick D. and Radcliffe, N.J., "Formal Algorithms + Formal Representations = Search Strategies", Parallel Problem Solving from Nature IV, Springer-Verlag LNCS 1141, pp 366-375, 1996.
 
 
                                            \[\arg\underset{S\in L}{\mathop{\min g}}\,\],
 
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 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 \[\left\{ {{r}^{(i)}} \right\}\] 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 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)
 
·        \[\Theta _{s}^{(i)}\in X_{s}^{(i)}\] (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, genetic operators, and parameters as sequences.
 
 
 
 
 
The execution of a GA typically begins by randomly sampling with replacement from Al.  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 (μ) is defined as the population size.
 
 
 
Adating he work of Lamont and Merkle (Lamont, 1997) we can define the termination criteria and the other genetic operators (GOs) 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 and the fitness scaling function Ts 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 P(t) into a new population 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 (PT). If$T(P)={P}'$, then P is a parent population and P/ is the offspring population. If$\mu ={\mu }'$, then it is called simply the population size.
 
 
 
The PT resulting from an GO often depends on the outcome of a random experiment. This result is referred to as a random population transformation (RPT or random PT). To define 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 an RPT. The distribution of PTs resulting from the application of a GO depends on the operator parameters; in other words, a GO maps its parameters to an RPT.
 
 
 
Now that both the fitness function and RPT have been defined, the GO can be defined in general: let$\mu \in {{\mathbb{Z}}^{+}}$, 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 $GOP\left( H,\mu ,X,\Omega  \right)$.
 
 
 
 
 
 
 
 
 
[1] Remember that 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 isnot necessarily surjective. The range of c determines the subset of Alavailable forexploration by the evolutionary algorithm.
 

Latest revision as of 06:10, 19 April 2013

2020 Mathematics Subject Classification: Primary: 90C99,60H25,68Q87 [MSN][ZBL]

$\newcommand{\argmin}[1]{\operatorname{argmin}_{#1}}$ Genetic algorithms (GAs) (also known as evolutionary algorithms [EAs]) assume a discrete search space $H$ and a function $f:H\to \R$. The general problem is to find $\argmin{x\in H}f$, 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 $c:A^l\to H$. In practice we may need a search space $\Xi\subseteq A^l$ to reflect the fact that some strings in the image $A^l$ under $c$ may represent invalid solutions to the 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 values to alleles. This statement of genes and alleles is often referred to as genotype-phenotype mapping. Given the statements above, the optimization becomes one of finding $\argmin{S\in L}$, where the function $g(s)=f(c(s))$. 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 $x$.

The execution of an GA typically begins by randomly 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 initialization, execution proceeds iteratively. Each iteration consists of an application of one or more of the genetic operators (GOs): recombination, mutation and selection. The combined effect of the GOs applied in a particular generation $t\in N$ is to transform the current population $P(t)$ into a new population $P(t+1)$. In the population transformation $\mu$, $\mu'\in\Z^+$ (the parent and offspring population sizes, respectively). A mapping $T:H^\mu\to H^{\mu'}$ is called a PT. If $T(P)=P'$ then $P$ is a parent population and $P'$ is the offspring population. If $\mu=\mu'$ it is simply the population size. The PT resulting from a GO often depends on the outcome of a random experiment. In Lamont and Merkel this result is referred to as a random population transformation (RPT or random PT). To define RPT, let $\mu \in \Z^+$and $\Omega $ be a set (the sample space). A random function $R:\Omega \to T(H^\mu,\bigcup\limits_{\mu'\in\Z^+} H^{\mu'})$ is called a RPT. The distribution of PTs resulting from the application of a GO depends on the operator parameters. In other words, a GO maps its parameters to an RPT. Since H is a nonempty set where $c:A^l\to H$, and $f:H\to\R$, the fitness scaling function can be defined as $T_s:\R\to\R$ and a related fitness function $\Phi \triangleq T_s\circ f\circ c$. In this definition it is understood that the objective function $f$ is determined by the application.

Now that both the fitness function and RPT have been defined, the GOs can be defined in general: let $\mu \in \Z^+$, $\aleph$ be a set (the parameter space) and $\Omega $ a set (the sample space). The mapping $X:\aleph \to T\left( \Omega ,T\left[ H^\mu,\bigcup\limits_{\mu'\in \Z^+} H^{\mu'} \right] \right)$ is a GO. The set of GOs is denoted as $GOP( H,\mu ,\aleph ,\Omega)$. 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.” In defining the GOs, we will start with recombination. Let $r\in GOP( H,\mu ,\aleph ,\Omega)$. If there exist $P\in H^\mu$, $\Theta \in \aleph $ and $\omega\in \Omega $ such that one individual in the offspring population $r_\Theta (P)$ depends on more than one individual of $P$ then $r$ is referred to as a recombination operator. A mutation is defined in the following manner: let $m\in GOP( H,\mu ,\aleph ,\Omega)$. 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 $m_\Theta(P)$ depends on at most one individual of $P$ then $m$ is called a mutation operator. Finally, for a selection operator let $s\in GOP( H,\mu ,\aleph \times T(H,\R),\Omega ) )$. If $P\in H^\mu$, $\Theta \in \aleph $, $f:H\to \R $ in all cases, and $s$ satisfies $a\in s_{( \Theta,\Phi)}(P) \Rightarrow a\in P$, then $s$ is a selection operator.


References

[1] Back, T., Evolutionary Algorithms in Theory and Practice, Oxford University Press, 1996

[2] Coello Coello, C.A., Van Veldhuizen, D.A. and Lamont, G.B., Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, 2002

[3] Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley Professional, 1989

[4] Holland, J.H., Adaptation in Natural and Artificial Systems, A Bradford Book, 1992

[5] Lamont, L. and Merkle, J.D., "A Random Based Framework for Evolutionary Algorithms", in: T. Back, Proceedings of the Seventh International Conference on Genetic Algorithms, 1997.

[6] Surry, Patrick D. and Radcliffe, N.J., "Formal Algorithms + Formal Representations = Search Strategies", Parallel Problem Solving from Nature IV, Springer-Verlag LNCS 1141, pp 366-375, 1996.

How to Cite This Entry:
Genetic Algorithms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genetic_Algorithms&oldid=27547