Editing Crossover (evolutionary algorithm) (section)

==Crossover for integer or real-valued genomes==
[[File:Discrete Recombination-3Dim.png|thumb|242x242px|Example of a discrete recombination in the three-dimensional case. The two possible offspring lie on the corners of the cuboid marked in blue.]]
For the crossover operators presented above and for most other crossover operators for bit strings, it holds that they can also be applied accordingly to integer or real-valued genomes whose genes each consist of an integer or real-valued number. Instead of individual bits, integer or real-valued numbers are then simply copied into the child genome. The offspring lie on the remaining corners of the hyperbody spanned by the two parents <math>P_1=(1.5, 6, 8)</math> and <math>P_2=(7, 2, 1)</math>, as exemplified in the accompanying image for the three-dimensional case. 

=== Discrete recombination ===
If the rules of the uniform crossover for bit strings are applied during the generation of the offspring, this is also called ''discrete recombination''.<ref name=":6">{{Cite book |last1=Eiben |first1=A.E. |url=http://link.springer.com/10.1007/978-3-662-44874-8 |title=Introduction to Evolutionary Computing |last2=Smith |first2=J.E. |date=2015 |publisher=Springer |isbn=978-3-662-44873-1 |edition=2nd |series=Natural Computing Series |location=Berlin, Heidelberg |pages=65–67 |chapter=Recombination Operators for Real-Valued Representation |doi=10.1007/978-3-662-44874-8 |s2cid=20912932}}</ref>

=== Intermediate recombination ===
[[File:Intermediate Recombination.png|thumb|242x242px|In the two-dimensional case, the two offspring of discrete recombination lie on the corners marked in blue, while the entire gray area is in question for the offspring of intermediate recombination.]]
In this recombination operator, the allele values of the child genome <math>a_i</math> are generated by mixing the alleles of the two parent genomes <math>a_{i,P_1}</math> and <math>a_{i,P_2}</math>:<ref name=":6" /><ref>{{Cite book |last1=Yu |first1=Xinjie |title=Introduction to Evolutionary Algorithms |last2=Gen |first2=Mitsuo |date=2010 |publisher=Springer |isbn=978-1-84996-128-8 |series=Decision Engineering |volume= |location=London |pages=45–63 |language=en |chapter=Real Code and Related Operators |doi=10.1007/978-1-84996-129-5}}</ref>
:<math>\alpha_i = \alpha_{i,P_1}\cdot\beta_i + \alpha_{i,P_2}\cdot\left (1 - \beta_i\right) \quad \mathsf{with} \quad  \beta_i \in \left [ -d, 1+d \right ]</math> randomly equally distributed per gene <math>i</math>
The choice of the interval <math>[-d, 1+d]</math> causes that besides the interior of the hyperbody spanned by the allele values of the parent genes additionally a certain environment for the range of values of the offspring is in question. A value of <math>0.25</math> is recommended for <math>d</math> to counteract the tendency to reduce the allele values that otherwise exists at <math>d=0</math>.<ref>{{Cite journal |last1=Mühlenbein |first1=Heinz |last2=Schlierkamp-Voosen |first2=Dirk |date=1993 |title=Predictive Models for the Breeder Genetic Algorithm I. Continuous Parameter Optimization |url=https://direct.mit.edu/evco/article/1/1/25-49/1093 |journal=Evolutionary Computation |language=en |volume=1 |issue=1 |pages=25–49 |doi=10.1162/evco.1993.1.1.25 |s2cid=16085506 |issn=1063-6560|url-access=subscription }}</ref> 

The adjacent figure shows for the two-dimensional case the range of possible new alleles of the two exemplary parents <math>P_1=(3,6)</math> and <math>P_2=(9,2)</math> in intermediate recombination. The offspring of discrete recombination <math>C_1</math> and <math>C_2</math> are also plotted. Intermediate recombination satisfies the arithmetic calculation of the allele values of the child genome required by virtual alphabet theory.<ref>{{Cite journal |last=Goldberg |first=David E. |date=1991 |title=Real-coded Genetic Algorithms, Virtual Alphabets, and Blocking |url=https://www.complex-systems.com/abstracts/v05_i02_a02/ |journal=Complex Syst. |volume=5 |issue=2 |pages=139–167}}</ref><ref>{{Cite book |last1=Stender |first1=J. |url= |title=Genetic algorithms in optimisation, simulation, and modelling |last2=Hillebrand |first2=E. |last3=Kingdon |first3=J. |date=1994 |publisher=IOS Press |isbn=90-5199-180-0 |location=Amsterdam |language=en |oclc=47216370}}</ref> Discrete and intermediate recombination are used as a standard in the [[evolution strategy]].<ref>{{Cite book |last=Schwefel |first=Hans-Paul |url=https://ls11-www.cs.tu-dortmund.de/lehre/wiley/ |title=Evolution and optimum seeking |date=1995 |publisher=Wiley |isbn=0-471-57148-2 |location=New York |language=en |oclc=30701094}}</ref>