Namespaces
Variants
Actions

Difference between revisions of "Genetic Algorithms"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Genetic Algorithms: Form and Function)
 
Line 5: Line 5:
 
1.      Genetic algorithms (GAs): basic form
 
1.      Genetic algorithms (GAs): basic form
  
A generic GA (aka evolutionary algorithm [EA]) assumes a discrete search space H and a function
+
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}.
+
where H is a subsetof the Euclidean space\mathbb{R}.
 
The general problem is to find
 
The general problem is to find
                                          \arg\underset{X\in H}{\mathop{\min }}\,f,
+
                                          \arg\underset{X\in H}{\mathop{\min }}\,f,
 
where X is avector of the decision variables and f is the objective function.
 
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 isrepresented by a string (or chromosome) s of length l madeup of symbols drawn from an alphabet Ausing the mapping
+
 
                                          c:{{A}^{l}}\toH.
+
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 Ausing the mapping
 +
                                          c:{{A}^{l}}\toH.
 
The mapping c is not necessarily surjective. The range of c determine the subset of Al  available for exploration by a GA.
 
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  
 
The range of c, Ξ  
                                          \Xi\subseteq {{A}^{l}}
+
                                          \Xi\subseteq {{A}^{l}}
isneeded to account for the fact that some strings in the image Al under c may represent invalid solutions to the original problem.
+
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.
  
The string length l depends on thedimensions 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:
 
Given the statements above, the optimization becomes:
                                \arg\underset{S\in L}{\mathop{\min g}}\,,
+
 
 +
                                  \arg\underset{S\in L}{\mathop{\min g}}\,,
 
given the function  
 
given the function  
                                                                                                g(s)=f(c(s)).
+
 
 +
                                  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.  
 
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.  
1.      Genetic algorithms and their operators
 
The following statementsabout the operators of Gas is adopted from Coello et al.  (Coello, 2002).
 
·        Let H be a nonempty set (the individual orsearch space), {{\left\{ {{u}^{i}} \right\}}_{i\in \mathbb{N}}} asequence in {{\mathbb{Z}}^{+}} (the parent populations),
 
·        {{\left\{ {{u}^{'(i)}} \right\}}_{i\in\mathbb{N}}} a sequence in {{\mathbb{Z}}^{+}}(the offspring populationsizes)
 
·        \phi :H\to \mathbb{R} a fitness function, \iota:\cup _{i=1}^{\infty }{{({{H}^{u}})}^{(i)}}\to {true, false} (thetermination criteria)
 
·        \chi \in {true, false}, r a sequence \left\{ {{r}^{(i)}} \right\} of recombinationoperators τ(i) : \[X_{r}^{(i)}\toT(\Omega _{r}^{(i)}
 
·        m asequence of {m(i)} ofmutation 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 recombinationparameters)
 
·        \Theta _{m}^{(i)}\in X_{m}^{(i)} (themutation parameters)
 
·        \Theta _{s}^{(i)}\in X_{s}^{(i)} (theselection parameters)
 
  
+
 
 +
2.      Genetic algorithms and their operators
 +
The following statements about the operators of GAs are adopted 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)}\toT(\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)}\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 byT:{{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.
 
Coello et al. define the collection μ (thenumber of individuals) via Hμ.The population transforms are denoted byT:{{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.

Revision as of 14:17, 14 August 2012

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 Ausing the mapping

                                          \[c:{{A}^{l}}\toH\].

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 adopted 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)}\toT(\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)}\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 byT:{{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.

How to Cite This Entry:
Genetic Algorithms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Genetic_Algorithms&oldid=27537