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,λ）EA 是有效的，标准 EA 最著名的运行时上限是氧（n25）。

我们证明一个（1,λ）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nη为了η≈3.97677（最多为次多项式因子）。因此，自我调节（1,λ）EA 的性能至少优于最佳固定参数n2.9767。

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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,λ) EA is effective, and for which the best known upper runtime bound for standard EAs is O(n25).

We prove that a (1,λ) EA self-adjusting the offspring population size λ using success-based rules optimises Cliff in O(n) expected generations and O(nlog⁡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 nη for η≈3.97677 (up to sub-polynomial factors). Hence, the self-adjusting (1,λ) EA outperforms the best fixed parameter by a factor of at least n2.9767.