Evolution strategy

The 'evolution strategy' optimization technique was created in the early 1960s and developed further in the 1970s and later by Ingo Rechenberg, Hans-Paul Schwefel and their co-workers.

Evolution strategies use natural problem-dependent representations, so problem space and 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,{{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}}{{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 }}{{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 }} pointing out that there are many variants.{{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.}}{{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 }}{{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 }}{{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 }} The real-valued chromosome contains, in addition to the $n$ decision variables, $n\text{'}$ mutation step sizes $\{\backslash sigma\}\_j$, where: $1\backslash leq\; j\backslash leq\; n\text{'}\backslash leq\; n$. Often one mutation step size is used for all decision variables or each has its own step size. Mate selection to produce $\backslash lambda$ offspring is random, i.e. independent of fitness. First, new mutation step sizes are generated per mating by intermediate recombination of the parental $\{\backslash sigma\; \}\_\{j\}$ with subsequent mutation as follows:

: $\{\backslash sigma\}\text{'}\_j\; =\; \backslash sigma\_j\; \backslash cdot\; e^\{(\backslash mathcal\{N\}(0,1)-\backslash mathcal\{N\}\_j(0,1))\}$

where $\backslash mathcal\{N\}(0,1)$ is a normally distributed random variable with mean $0$ and standard deviation $1$. $\backslash mathcal\{N\}(0,1)$ applies to all $\{\backslash sigma\}\text{'}\_j$, while $\backslash mathcal\{N\}\_j(0,1)$ is newly determined for each $\{\backslash sigma\}\text{'}\_j$. Next, discrete recombination of the decision variables is followed by a mutation using the new mutation step sizes as standard deviations of the normal distribution. The new decision variables $x\_j\text{'}$ are calculated as follows:

:$x\_j\text{'}=x\_j+\backslash mathcal\{N\}\_j(0,\{\backslash sigma\}\_j\text{'})$

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.

The ES knows two variants of best selection for the generation of the next parent population ($\backslash mu$ - number of parents, $\backslash lambda$ - number of offspring):

• $(\backslash mu,\backslash lambda\; )$: The $\backslash mu$ best offspring are used for the next generation (usually $\backslash mu=\backslash frac\{\backslash lambda\}\{2\}$).
• $(\backslash mu\; +\backslash lambda\; )$: The best are selected from a union of $\backslash mu$ parents and $\backslash lambda$ offspring.

Bäck and Schwefel recommend that the value of $\backslash lambda$ should be approximately seven times the $\backslash mu$, whereby $\backslash mu$ must not be chosen too small because of the strong selection pressure. Suitable values for $\backslash mu$ are application-dependent and must be determined experimentally. The selection of the next generation in evolution strategies is deterministic and only based on the fitness rankings, not on the actual fitness values. The resulting algorithm is therefore invariant with respect to monotonic transformations of the objective function.

The simplest and oldest evolution strategy $\backslash mathit\{(1+1)\}$ operates on a population of size two: the current point (parent) and the result of its mutation. Only if the mutant's fitness is at least as good as the parent one, it becomes the parent of the next generation. Otherwise the mutant is disregarded. More generally, $\backslash lambda$ mutants can be generated and compete with the parent, called $(1+\backslash lambda\; )$. In $(1,\backslash lambda\; )$ the best mutant becomes the parent of the next generation while the current parent is always disregarded. For some of these variants, proofs of linear convergence (in a stochastic sense) have been derived on unimodal objective functions.{{cite journal |last1=Auger |first1=Anne |title=Convergence results for the ( 1 , λ ) -SA-ES using the theory of ϕ -irreducible Markov chains |journal=Theoretical Computer Science |date=April 2005 |volume=334 |issue=1–3 |pages=35–69 |doi=10.1016/j.tcs.2004.11.017}}{{cite journal |last1=Jägersküpper |first1=Jens |title=How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms |journal=Theoretical Computer Science |date=August 2006 |volume=361 |issue=1 |pages=38–56 |doi=10.1016/j.tcs.2006.04.004}}

Individual step sizes for each coordinate, or correlations between coordinates, which are essentially defined by an underlying covariance matrix, are controlled in practice either by self-adaptation or by covariance matrix adaptation (CMA-ES). When the mutation step is drawn from a multivariate normal distribution using an evolving covariance matrix, it has been hypothesized that this adapted matrix approximates the inverse Hessian of the search landscape. This hypothesis has been proven for a static model relying on a quadratic approximation.{{cite journal |last1=Shir |first1=Ofer M. |last2=Yehudayoff |first2=Amir |title=On the covariance-Hessian relation in evolution strategies |journal=Theoretical Computer Science |date=January 2020 |volume=801 |pages=157–174 |doi=10.1016/j.tcs.2019.09.002|arxiv=1806.03674 }} In 2025, Chen et.al.{{Cite journal |last1=Chen |first1=Tai-You |last2=Chen |first2=Wei-Neng |last3=Hao |first3=Jin-Kao |last4=Wang |first4=Yang |last5=Zhang |first5=Jun |date=2025 |title=Multi-Agent Evolution Strategy With Cooperative and Cumulative Step Adaptation for Black-Box Distributed Optimization |url=https://ieeexplore.ieee.org/document/10824905 |journal=IEEE Transactions on Evolutionary Computation |pages=1 |doi=10.1109/TEVC.2025.3525713 |issn=1941-0026|url-access=subscription }} proposed a multi-agent evolution strategy for consensus-based distributed optimization, where a novel step adaptation method is designed to help multiple agents control the step size cooperatively.

• Covariance matrix adaptation evolution strategy (CMA-ES)
• Derivative-free optimization
• Evolutionary computation
• Genetic algorithm
• Natural evolution strategy
• Evolutionary game theory

{{Reflist}}

• Ingo Rechenberg (1971): Evolutionsstrategie – Optimierung technischer Systeme nach Prinzipien der biologischen Evolution (PhD thesis). Reprinted by Frommann-Holzboog (1973). {{ISBN|3-7728-1642-8}}
• Hans-Paul Schwefel (1974): Numerische Optimierung von Computer-Modellen (PhD thesis). Reprinted by Birkhäuser (1977).
• Hans-Paul Schwefel: [https://www.researchgate.net/publication/220690578_Evolution_and_Optimum_Seeking Evolution and Optimum Seeking]. New York: Wiley & Sons 1995. {{ISBN|0-471-57148-2}}
• H.-G. Beyer and H.-P. Schwefel. [https://journals.openedition.org/trivium/3664 Evolution Strategies: A Comprehensive Introduction]. Journal Natural Computing, 1(1):3–52, 2002.
• Hans-Georg Beyer: The Theory of Evolution Strategies. Springer, April 27, 2001. {{ISBN|3-540-67297-4}}
• Ingo Rechenberg: Evolutionsstrategie '94. Stuttgart: Frommann-Holzboog 1994. {{ISBN|3-7728-1642-8}}
• J. Klockgether and H. P. Schwefel (1970). [https://www.researchgate.net/profile/Hans-Paul_Schwefel/publication/236373493_TWO-PHASE_NOZZLE_AND_HOLLOW_CORE_JET_EXPERIMENTS/links/544bd4db0cf2d6347f43a164.pdf Two-Phase Nozzle And Hollow Core Jet Experiments]. AEG-Forschungsinstitut. MDH Staustrahlrohr Project Group. Berlin, Federal Republic of Germany. Proceedings of the 11th Symposium on Engineering Aspects of Magneto-Hydrodynamics, Caltech, Pasadena, Cal., 24.–26.3. 1970.
• M. Emmerich, O.M. Shir, and H. Wang: [https://link.springer.com/referenceworkentry/10.1007/978-3-319-07153-4_13-1 Evolution Strategies]. In: Handbook of Heuristics, 1-31. Springer International Publishing (2018).

• [https://web.archive.org/web/20180425010001/http://www.bionik.tu-berlin.de/institut/xstart.htm Bionics & Evolutiontechnique at Technische Universität Berlin]
• [http://ls11-www.cs.uni-dortmund.de/ Chair of Algorithm Engineering (Ls11) – TU Dortmund University]
• [https://archive.today/20130106090846/http://sfbci.cs.uni-dortmund.de/ Collaborative Research Center 531 – TU Dortmund University]

{{Evolutionary computation}}

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