Optimal Static and Self-Adjusting Parameter Choices for the ( 1 + ( λ , λ ) ) Genetic Algorithm

Abstract : The (1 + (λ, λ)) genetic algorithm (GA) proposed in [Doerr, Doerr, and Ebel. From black-box complexity to designing new genetic algorithms. Theoretical Computer Science (2015)] is one of the few examples for which a super-constant speed-up of the expected optimization time through the use of crossover could be rigorously demonstrated. It was proven that the expected optimization time of this algorithm on OneMax is O(max{n log(n)/λ, λn}) for any offspring population size λ ∈ {1,. .. , n} (and the other parameters suitably dependent on λ) and it was shown experimentally that a self-adjusting choice of λ leads to a better, most likely linear, runtime. In this work, we study more precisely how the optimization time depends on the parameter choices, leading to the following results on how to optimally choose the population size, the mutation probability, and the crossover bias both in a static and a dynamic fashion. For the mutation probability and the crossover bias depending on λ as in [DDE15], we improve the previous runtime bound to O(max{n log(n)/λ, nλ log log(λ)/ log(λ)}). This expression is minimized by a value of λ slightly larger than what the previous result suggested and gives an expected optimization time of O n log(n) log log log(n)/ log log(n). We show that no static choice in the three-dimensional parameter space of offspring population, mutation probability, and crossover bias gives an asymp-totically better runtime. Results presented in this work are based on [12–14]. B. DoerrÉcole Doerr´DoerrÉcole Polytechnique, LIX-UMR 7161, We also prove that the self-adjusting parameter choice suggested in [DDE15] outperforms all static choices and yields the conjectured linear expected runtime. This is asymptotically optimal among all possible parameter choices.
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Benjamin Doerr, Carola Doerr. Optimal Static and Self-Adjusting Parameter Choices for the ( 1 + ( λ , λ ) ) Genetic Algorithm. Algorithmica, Springer Verlag, 2018, 80, pp.1658-1709. ⟨10.1007/s00453-017-0354-9⟩. ⟨hal-01668262⟩

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