Evolution strategy

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title: "Evolution strategy" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["evolution-strategy"] description: "Algorithm in computer science" topic_path: "science/biology" source: "https://en.wikipedia.org/wiki/Evolution_strategy" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Algorithm in computer science ::

Evolution strategy (ES) from computer science is a subclass of evolutionary algorithms, which serves as an optimization technique. It uses the major genetic operators mutation, recombination and selection of parents.

History

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. ::data[format=table title="Timeline of ES - selected algorithms"]

Year	Description	Reference
1973	ES introduced with mutation and selection
1994	Derandomized self-adaptation ES - Derandomized scheme of mutative step size control is used
1994	CSA-ES - usage information from the old generations
2001	CMA-ES
2006	Weighted multi-recombination ES - usage of weighted recombination
2007	Meta-ES - incremental aggregation of partial semantic structures
2008	Natural ES - usage of natural gradient
2010	Exponential natural ES - a simpler version of natural ES
2014	Limited memory CMA-ES - time–memory complexity reduction by covariance matrix decomposition
2016	Fitness inheritance CMA-ES - fitness evaluation computational cost reduction using fitness inheritance
2017	RS-CMSA ES - usage of subpopulations
2017	MA-ES - COV update and COV matrix square root are not used
2018	Weighted ES - weighted recombination of general convex quadratic functions
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Methods

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, pointing out that there are many variants. The real-valued chromosome contains, in addition to the n decision variables, n' mutation step sizes {\sigma}j, where: 1\leq j\leq n'\leq n. Often one mutation step size is used for all decision variables or each has its own step size. Mate selection to produce \lambda offspring is random, i.e. independent of fitness. First, new mutation step sizes are generated per mating by intermediate recombination of the parental {\sigma }{j} with subsequent mutation as follows: : {\sigma}'_j = \sigma_j \cdot e^{(\mathcal{N}(0,1)-\mathcal{N}_j(0,1))} where \mathcal{N}(0,1) is a normally distributed random variable with mean 0 and standard deviation 1. \mathcal{N}(0,1) applies to all {\sigma}'_j, while \mathcal{N}_j(0,1) is newly determined for each {\sigma}'_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' are calculated as follows: :x_j'=x_j+\mathcal{N}_j(0,{\sigma}_j') 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.

Variants

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

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

Bäck and Schwefel recommend that the value of \lambda should be approximately seven times the \mu, whereby \mu must not be chosen too small because of the strong selection pressure. Suitable values for \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

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. In 2025, Chen et.al. 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.

References

Bibliography

• Ingo Rechenberg (1971): Evolutionsstrategie – Optimierung technischer Systeme nach Prinzipien der biologischen Evolution (PhD thesis). Reprinted by Frommann-Holzboog (1973).
• Hans-Paul Schwefel (1974): Numerische Optimierung von Computer-Modellen (PhD thesis). Reprinted by Birkhäuser (1977).
• Hans-Paul Schwefel: Evolution and Optimum Seeking. New York: Wiley & Sons 1995.
• H.-G. Beyer and H.-P. Schwefel. 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.
• Ingo Rechenberg: Evolutionsstrategie '94. Stuttgart: Frommann-Holzboog 1994.
• J. Klockgether and H. P. Schwefel (1970). 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: Evolution Strategies. In: Handbook of Heuristics, 1-31. Springer International Publishing (2018).

Research centers

• Bionics & Evolutiontechnique at Technische Universität Berlin
• Chair of Algorithm Engineering (Ls11) – TU Dortmund University
• Collaborative Research Center 531 – TU Dortmund University

References

1. (1 August 2020). "Evolutionary algorithms and their applications to engineering problems". Neural Computing and Applications.
2. (1 June 2019). "(μ+λ) Evolution strategy algorithm in well placement, trajectory, control and joint optimisation". Journal of Petroleum Science and Engineering.
3. (January 1975). "Rechenberg, Ingo, Evolutionsstrategie — Optimierung technischer Systeme nach Prinzipien der biologischen Evolution. 170 S. mit 36 Abb. Frommann-Holzboog-Verlag. Stuttgart 1973. Broschiert". Feddes Repertorium.
4. (December 1994). "A Derandomized Approach to Self-Adaptation of Evolution Strategies". Evolutionary Computation.
5. (1994). "Parallel Problem Solving from Nature — PPSN III". Springer.
6. (June 2001). "Completely Derandomized Self-Adaptation in Evolution Strategies". Evolutionary Computation.
7. (28 August 2006). "Weighted multirecombination evolution strategies". Theoretical Computer Science.
8. (2007). "Artificial Neural Networks – ICANN 2007". Springer.
9. (1 January 2014). "Natural evolution strategies". J. Mach. Learn. Res..
10. (7 July 2010). "Proceedings of the 12th annual conference on Genetic and evolutionary computation". Association for Computing Machinery.
11. (12 July 2014). "Proceedings of the 2014 Annual Conference on Genetic and Evolutionary Computation". Association for Computing Machinery.
12. (July 2016). "2016 IEEE Congress on Evolutionary Computation (CEC)".
13. (September 2017). "Multimodal Optimization by Covariance Matrix Self-Adaptation Evolution Strategy with Repelling Subpopulations". Evolutionary Computation.
14. (October 2017). "Simplify Your Covariance Matrix Adaptation Evolution Strategy". IEEE Transactions on Evolutionary Computation.
15. (6 September 2020). "Quality gain analysis of the weighted recombination evolution strategy on general convex quadratic functions". Theoretical Computer Science.
16. Schwefel, Hans-Paul. (1995). "Evolution and Optimum Seeking". Wiley.
17. (1993). "An Overview of Evolutionary Algorithms for Parameter Optimization". Evolutionary Computation.
18. (1995). "Contemporary Evolution Strategies". Springer.
19. (1991). "A Survey of Evolution Strategies". Morgan Kaufmann.
20. (2016). "Hybrid Self-Adaptive Evolution Strategies Guided by Neighborhood Structures for Combinatorial Optimization Problems". Evolutionary Computation.
21. (2001). "Completely Derandomized Self-Adaptation in Evolution Strategies". Evolutionary Computation.
22. (2004). "Evaluating the CMA Evolution Strategy on Multimodal Test Functions". Springer.
23. (April 2005). "Convergence results for the ( 1 , λ ) -SA-ES using the theory of ϕ -irreducible Markov chains". Theoretical Computer Science.
24. (August 2006). "How the (1+1) ES using isotropic mutations minimizes positive definite quadratic forms". Theoretical Computer Science.
25. (January 2020). "On the covariance-Hessian relation in evolution strategies". Theoretical Computer Science.
26. (2025). "Multi-Agent Evolution Strategy With Cooperative and Cumulative Step Adaptation for Black-Box Distributed Optimization". IEEE Transactions on Evolutionary Computation.

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