Understanding when evolutionary algorithms are efficient or not, and how they efficiently solve problems, is one of the central research tasks in evolutionary computation. In this work, we make progress in understanding the interplay between parent and offspring population size of the (µ, λ) EA. Previous works, roughly speaking, indicate that for λ ≥ (1 + ε)eµ, this EA easily optimizes the OneMax function, whereas an offspring population size λ ≤ (1 − ε)eµ leads to an exponential runtime.Motivated also by the observation that in the efficient regime the (µ, λ) EA loses its ability to escape local optima, we take a closer look into this phase transition. Among other results, we show that when µ ≤ n 1/2−c for any constant c > 0, then for any λ ≤ eµ we have a super-polynomial runtime. However, if µ ≥ n 2/3+c , then for any λ ≥ eµ, the runtime is polynomial. For the latter result we observe that the (µ, λ) EA profits from better individuals also because these, by creating slightly worse offspring, stabilize slightly sub-optimal subpopulations. While these first results close to the phase transition

Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of true population-based evolutionary algorithms remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the (µ + λ) evolutionary algorithm on the simple OneMax benchmark function, only the special cases µ = 1 and λ = 1 have been solved.In this work, we analyze this long-standing problem and show the asymptotically tight result that the runtime T , the number of iterations until the optimum is found, satisfieswhere log + x := max{1, log x} for all x > 0. The same methods allow to improve the previous-best O( n log n λ + n log λ) runtime guarantee for the (λ+λ) EA with fair parent selection to a tight Θ( n log n λ + n) runtime result. * A preliminary version of this work [ADFH18] was presented at the Genetic and Evolutionary Computation Conference (GECCO) 2018. In this version, the presentation was improved by rewriting almost the entire text, by giving a clearer comparison with the previous state of the art, by making many proofs more rigorous, by extending the lower bounds to arbitrary fitness functions (subject to a mild restriction on the number of global optima), and by extending our results to the so-called (N + N ) EA using a fair parent selection.

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