Editing Evolution strategy (section)

==Methods==
Evolution strategies use natural problem-dependent representations, so [[Genetic representation|problem space]] and [[Genetic representation|search space]] are identical. In common with [[evolutionary algorithms]], the operators are applied in a loop. An iteration of the loop is called a generation. The sequence of generations is continued until a termination criterion is met.

The special feature of the ES is the self-adaptation of mutation step sizes and the [[coevolution]] associated with it. The ES is briefly presented using the standard form,<ref>{{Cite book |last=Schwefel |first=Hans-Paul |title=Evolution and Optimum Seeking |date=1995 |publisher=Wiley |isbn=978-0-471-57148-3 |edition= |series=Sixth-generation computer technology series |location=New York |url=https://www.researchgate.net/publication/220690578}}</ref><ref name=":0">{{Cite journal |last1=Bäck |first1=Thomas |last2=Schwefel |first2=Hans-Paul |date=1993 |title=An Overview of Evolutionary Algorithms for Parameter Optimization |url=https://direct.mit.edu/evco/article/1/1/1-23/1092 |journal=Evolutionary Computation |language=en |volume=1 |issue=1 |pages=1–23 |doi=10.1162/evco.1993.1.1.1 |issn=1063-6560|url-access=subscription }}</ref><ref>{{Citation |last1=Schwefel |first1=Hans-Paul |last2=Rudolph |first2=Günter |last3=Bäck |first3=Thomas |title=Contemporary Evolution Strategies |date=1995 |url=https://www.researchgate.net/publication/2606149 |work=Proceedings of the Third European Conference on Advances in Artificial Life |pages=893–907 |place=Berlin, Heidelberg |publisher=Springer |isbn=978-3-540-59496-3 |doi=10.1007/3-540-59496-5_351 |editor-last1=Morán |editor-first1=Frederico |editor-last2=Moreno |editor-first2=Alvaro |editor-last3=Merelo |editor-first3=J.J. |editor-last4=Chacón |editor-first4=Pablo }}</ref> pointing out that there are many variants.<ref>{{Citation |last1=Bäck |first1=Thomas |title=A Survey of Evolution Strategies |date=1991 |work=Proceedings of the Fourth International Conference on Genetic Algorithms (ICGA) |editor-last1=Belew |editor-first1=Richard K. |location=San Mateo, Calif |publisher=Morgan Kaufmann |isbn=978-1-55860-208-3 |last2=Hoffmeister |first2=Frank |last3=Schwefel |first3=Hans-Paul |editor-last2=Booker |editor-first2=Lashon B.}}</ref><ref>{{Cite journal |last1=Coelho |first1=V. N. |last2=Coelho |first2=I. M. |last3=Souza |first3=M. J. F. |last4=Oliveira |first4=T. A. |last5=Cota |first5=L. P. |last6=Haddad |first6=M. N. |last7=Mladenovic |first7=N. |last8=Silva |first8=R. C. P. |last9=Guimarães |first9=F. G. |date=2016 |title=Hybrid Self-Adaptive Evolution Strategies Guided by Neighborhood Structures for Combinatorial Optimization Problems |url=https://direct.mit.edu/evco/article/24/4/637-666/1033 |journal=Evolutionary Computation |language=en |volume=24 |issue=4 |pages=637–666 |doi=10.1162/EVCO_a_00187 |pmid=27258842 |issn=1063-6560|url-access=subscription }}</ref><ref name=":1">{{Cite journal |last1=Hansen |first1=Nikolaus |last2=Ostermeier |first2=Andreas |date=2001 |title=Completely Derandomized Self-Adaptation in Evolution Strategies |url=https://direct.mit.edu/evco/article/9/2/159-195/892 |journal=Evolutionary Computation |language=en |volume=9 |issue=2 |pages=159–195 |doi=10.1162/106365601750190398 |pmid=11382355 |issn=1063-6560|url-access=subscription }}</ref><ref>{{Citation |last1=Hansen |first1=Nikolaus |title=Evaluating the CMA Evolution Strategy on Multimodal Test Functions |date=2004 |url=http://link.springer.com/10.1007/978-3-540-30217-9_29 |work=Parallel Problem Solving from Nature - PPSN VIII |volume=3242 |pages=282–291 |editor-last=Yao |editor-first=Xin |access-date= |place=Berlin, Heidelberg |publisher=Springer |doi=10.1007/978-3-540-30217-9_29 |isbn=978-3-540-23092-2 |last2=Kern |first2=Stefan |editor2-last=Burke |editor2-first=Edmund K. |editor3-last=Lozano |editor3-first=José A. |editor4-last=Smith |editor4-first=Jim|url-access=subscription }}</ref> The real-valued chromosome contains, in addition to the <math>n</math> decision variables, <math>n'</math> mutation step sizes <math>{\sigma}_j</math>, where: <math>1\leq j\leq n'\leq n</math>. Often one mutation step size is used for all decision variables or each has its own step size. Mate selection to produce <math>\lambda</math> offspring is random, i.e. independent of fitness. First, new mutation step sizes are generated per mating by [[Crossover (genetic algorithm)#Intermediate recombination|intermediate recombination]] of the parental <math>{\sigma }_{j}</math> with subsequent mutation as follows:
: <math> {\sigma}'_j = \sigma_j \cdot e^{(\mathcal{N}(0,1)-\mathcal{N}_j(0,1))} </math>
where <math>\mathcal{N}(0,1)</math> is a [[Normal distribution|normally distributed]] random variable with mean <math>0</math> and standard deviation <math>1</math>. <math>\mathcal{N}(0,1)</math> applies to all <math>{\sigma}'_j</math>, while <math>\mathcal{N}_j(0,1)</math> is newly determined for each <math>{\sigma}'_j</math>. Next, [[Crossover (genetic algorithm)#Discrete recombination|discrete recombination]] of the decision variables is followed by a [[Mutation (genetic algorithm)#Mutation without consideration of restrictions|mutation]] using the new mutation step sizes as standard deviations of the normal distribution. The new decision variables <math> x_j' </math> are calculated as follows:
:<math>x_j'=x_j+\mathcal{N}_j(0,{\sigma}_j')</math> 
This results in an evolutionary search on two levels: First, at the problem level itself and second, at the mutation step size level. In this way, it can be ensured that the ES searches for its target in ever finer steps. However, there is also the danger of being able to skip larger invalid areas in the search space only with difficulty.