In the discrete domain, self-adjusting parameters of evolutionary algorithms (EAs) have emerged as a fruitful research area with many runtime analyses showing that self-adjusting parameters can outperform the best fixed parameters. Most existing runtime analyses focus on elitist EAs on simple problems, for which moderate performance gains were shown. Here we consider a much more challenging scenario: the multimodal function Cliff, defined as an example where a $(1,\lambda )$ EA is effective, and for which the best known upper runtime bound for standard EAs is $O(n^{25})$.

We prove that a $(1,\lambda )$ EA self-adjusting the offspring population size λ using success-based rules optimises Cliff in $O(n)$ expected generations and $O(n\log n)$ expected evaluations. Along the way, we prove tight upper and lower bounds on the runtime for fixed λ (up to a logarithmic factor) and identify the runtime for the best fixed λ as $n^{\eta }$ for $\eta \approx 3.97677$ (up to sub-polynomial factors). Hence, the self-adjusting $(1,\lambda )$ EA outperforms the best fixed parameter by a factor of at least $n^{2.9767}$.

在离散领域，进化算法（EA）的自调整参数已成为一个富有成果的研究领域，许多运行时分析表明自调整参数可以优于最佳固定参数。大多数现有的运行时分析都集中在精英 EA 上解决简单问题，这些问题显示出适度的性能提升。在这里，我们考虑一个更具挑战性的场景：多模态函数 Cliff，定义为一个示例，其中$（1,\lambda ）$EA 是有效的，标准 EA 最著名的运行时上限是$氧（n^{25}）$。

我们证明一个$（1,\lambda ）$EA使用基于成功的规则自我调整后代种群大小λ ，从而优化 Cliff$氧（n）$预期世代和$氧（n\mathrm{日志}n）$预期评价。在此过程中，我们证明了固定λ （最多为对数因子）的运行时间的严格上限和下限，并将最佳固定λ的运行时间确定为$n^{\eta }$为了$\eta \approx 3.97677$（最多为次多项式因子）。因此，自我调节$（1,\lambda ）$EA 的性能至少优于最佳固定参数$n^{2.9767}$。