Optimal Static and Self-Adjusting Parameter Choices for the (1+(λ,λ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(

AbstractThe $$(1+(\lambda ,\lambda ))$$(1+(λ,λ)) genetic algorithm proposed in Doerr et al. (Theor Comput Sci 567:87–104, 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 $$O(\max \{n \log (n) / \lambda , \lambda n\})$$O(max{nlog(n)/λ,λn}) for any offspring population size $$\lambda \in \{1,\ldots ,n\}$$λ∈{1,…,n} (and the other parameters suitably dependent on $$\lambda $$λ) and it was shown experimentally that a self-adjusting choice of $$\lambda $$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 $$\lambda $$λ as in Doerr et al. (2015), we improve the previous runtime bound to $$O(\max \{n \log (n) / \lambda , n \lambda \log \log (\lambda )/\log (\lambda )\})$$O(max{nlog(n)/λ,nλloglog(λ)/log(λ)}). This expression is minimized by a value of $$\lambda $$λ slightly larger than what the previous result suggested and gives an expected optimization time of $$O\left( n \sqrt{\log (n) \log \log \log (n) / \log \log (n)}\right) $$Onlog(n)logloglog(n)/loglog(n). We show that no static choice in the three-dimensional parameter space of offspring population, mutation probability, and crossover bias gives an asymptotically better runtime. We also prove that the self-adjusting parameter choice suggested in Doerr et al. (2015) outperforms all static choices and yields the conjectured linear expected runtime. This is asymptotically optimal among all possible parameter choices.