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A Tight Runtime Analysis for the (µ + λ) EA

Abstract : Despite significant progress in the theory of evolutionary algorithms, the theoretical understanding of evolutionary algorithms which use non-trivial populations remains challenging and only few rigorous results exist. Already for the most basic problem, the determination of the asymptotic runtime of the $(\mu+\lambda)$ evolutionary algorithm on the simple OneMax benchmark function, only the special cases $\mu=1$ and $\lambda=1$ 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, satisfies \[E[T] = \Theta\bigg(\frac{n\log n}{\lambda}+\frac{n}{\lambda / \mu} + \frac{n\log^+\log^+ \lambda/ \mu}{\log^+ \lambda / \mu}\bigg),\] where $\log^+ x := \max\{1, \log x\}$ for all $x > 0$. The same methods allow to improve the previous-best $O(\frac{n \log n}{\lambda} + n \log \lambda)$ runtime guarantee for the $(\lambda+\lambda)$~EA with fair parent selection to a tight $\Theta(\frac{n \log n}{\lambda} + n)$ runtime result.
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https://hal.archives-ouvertes.fr/hal-03353025
Contributor : Benjamin Doerr Connect in order to contact the contributor
Submitted on : Thursday, September 23, 2021 - 4:35:44 PM
Last modification on : Friday, September 24, 2021 - 3:32:33 AM

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Denis Antipov, Benjamin Doerr. A Tight Runtime Analysis for the (µ + λ) EA. Algorithmica, Springer Verlag, 2021, 83 (4), pp.1054-1095. ⟨10.1007/s00453-020-00731-5⟩. ⟨hal-03353025⟩

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