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Evolutionary self-adaptation: a survey of operators and strategy parameters

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Abstract

The success of evolutionary search depends on adequate parameter settings. Ill conditioned strategy parameters decrease the success probabilities of genetic operators. Proper settings may change during the optimization process. The question arises if adequate settings can be found automatically during the optimization process. Evolution strategies gave an answer to the online parameter control problem decades ago: self-adaptation. Self-adaptation is the implicit search in the space of strategy parameters. The self-adaptive control of mutation strengths in evolution strategies turned out to be exceptionally successful. Nevertheless, for years self-adaptation has not achieved the attention it deserves. This paper is a survey of self-adaptive parameter control in evolutionary computation. It classifies self-adaptation in the taxonomy of parameter setting techniques, gives an overview of automatic online-controllable evolutionary operators and provides a coherent view on search techniques in the space of strategy parameters. Beyer and Sendhoff’s covariance matrix self-adaptation evolution strategy is reviewed as a successful example for self-adaptation and exemplarily tested for various concepts that are discussed.

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Notes

  1. Typically, genetic operators are crossover/recombination, mutation, inversion, gene deletion and duplication.

  2. According to a tradition in ES, μ is the size of the parental population, while λ is the size of the offspring population.

  3. Originally, σ denotes step sizes in ES. We use this symbol for all kinds of strategy variables in this paper.

  4. The concept of global multi-recombination will be explained in Sect. 4.2.

  5. For log-normal mutation of step sizes, see Sect. 4.1.2.

  6. The index j denotes the index of the j-th ranked individual of the λ offspring individuals with regard to fitness f(x j ).

  7. The Sphere problem is the continuous optimization problem to find an \({\bf x} \in {\mathbb{R}}^N\) minimizing \(f({\bf x}):=\sum_{i=1}^{N} x_{i}^{2}.\)

  8. Optimal progress is problem-dependent and can be stated theoretically on artificial functions [12].

  9. The success probability p s of an evolutionary operator is the relation of solutions with better fitness than the original solution f(x) and all generated solutions.

  10. OneMax is a pseudo-binary function. For a bit string \({\bf x} {\in}\{0,1\}^N\), maximize \(f({\bf x}):=\sum_{i=1}^{N}x_i \quad\hbox{ with } {\bf x} \in \{0,1\}^N\) optimum x * = (1, ..., 1) with f(x *) = N.

  11. 3-SAT is a propositional satisfiability problem, each clause contains k = 3 literals; we randomly generate formulas with N = 50 variables and 150 clauses; the fitness function is: \({\frac{\#\hbox{of true clauses}} {\#\hbox{of all clauses}}}.\)

  12. Schwefel’s problem 2.40 [69]: Minimize f(x): =  − ∑ 5i=1 x i , constraints \(g_{j}({\bf x}):=x_{j} {\geq}0, \hbox{for } j=1,\ldots,5\) and \(g_{j}({\bf x})=-\sum\nolimits^{5}_{i=1}(9+i)x_{i}+50000 {\geq}0, \hbox{for } j=6\), minimum x * = (5000, 0, 0, 0, 0)T with f(x *) =  − 5000.

  13. TR (tangent problem): minimize \(f({\bf x}):=\sum_{i=1}^{N} x_{i}^{2}\), constraints \(g({\bf x}):=\sum_{i=1}^{N} x_{i}-t>0, \qquad t \in {\mathbb{R}} \qquad \hbox{(tangent)},\) for N=k and t=k the minimum lies at: \({\bf x}^{*}=(1,\ldots,1)^{T},\hbox{with }\quad f({\bf x}^{*})=k\), TR2 means N = 2 and t = 2.

  14. difference between the optimum and the best solution \(|f({\bf x}^{*})-f({\bf x}^{best})|.\)

  15. Low innovation rates are caused by variation operators that produce offspring not far away from their parents, e.g. by low mutation rates.

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Acknowledgments

The author thanks Günter Rudolph and the anonymous reviewers for their helpful comments to improve the manuscript.

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    Oliver Kramer

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Kramer, O. Evolutionary self-adaptation: a survey of operators and strategy parameters. Evol. Intel. 3, 51–65 (2010). https://doi.org/10.1007/s12065-010-0035-y

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