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.

中文翻译：

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$(\mu + \lambda)$ EA 的严格运行时分析

尽管进化算法理论取得了重大进展，但对使用非平凡种群的进化算法的理论理解仍然具有挑战性，并且只有少数严格的结果存在。已经对于最基本的问题，在简单的 OneMax 基准函数上确定 $(\mu+\lambda)$ 进化算法的渐近运行时间，只有特殊情况 $\mu=1$ 和 $\lambda=1$ 有已解决。在这项工作中，我们分析了这个长期存在的问题，并展示了运行时间 $T$，即找到最优值之前的迭代次数，满足 \[E[T] = \Theta\bigg(\frac{ n\log n}{\lambda}+\frac{n}{\lambda / \mu} + \frac{n\log^+\log^+ \lambda/ \mu}{\log^+ \lambda / \ mu}\bigg),\] 其中 $\log^+ x := \max\{1, \log x\}$ 对于所有 $x > 0$。