It is generally accepted that populations are useful for the global exploration of multi-modal optimisation problems. Indeed, several theoretical results are available showing such advantages over single-trajectory search heuristics. In this paper we provide evidence that evolving populations via crossover and mutation may also benefit the optimisation time for hillclimbing unimodal functions. In particular, we prove bounds on the expected runtime of the standard (μ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu +1$$\end{document}) GA for OneMax that are lower than its unary black box complexity and decrease in the leading constant with the population size up to μ=ologn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =o\left( \sqrt{\log n}\right) $$\end{document}. Our analysis suggests that the optimal mutation strategy is to flip two bits most of the time. To achieve the results we provide two interesting contributions to the theory of randomised search heuristics: (1) A novel application of drift analysis which compares absorption times of different Markov chains without defining an explicit potential function. (2) The inversion of fundamental matrices to calculate the absorption times of the Markov chains. The latter strategy was previously proposed in the literature but to the best of our knowledge this is the first time is has been used to show non-trivial bounds on expected runtimes.

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论种群对标准稳态遗传算法开发速度的益处

人们普遍认为，种群对于多模态优化问题的全局探索很有用。事实上，有几个理论结果显示了这种优于单轨迹搜索启发式的优势。在本文中，我们提供的证据表明，通过交叉和突变进化种群也可能有益于爬山单峰函数的优化时间。特别是，我们证明了标准预期运行时间的界限 (μ+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{ mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu +1$$\end{document}) OneMax 的 GA 低于其一元黑盒复杂度并降低在人口规模高达 μ=ologn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs } \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =o\left( \sqrt{\log n}\right) $$\end{document}。我们的分析表明，最佳变异策略是在大多数情况下翻转两位。为了获得结果，我们对随机搜索启发式理论提供了两个有趣的贡献：（1）漂移分析的一种新应用，它比较了不同马尔可夫链的吸收时间，而无需定义明确的势函数。(2) 基本矩阵的求逆计算马尔可夫链的吸收时间。后一种策略以前在文献中提出过，但据我们所知，这是第一次用于显示预期运行时的非平凡边界。(2) 基本矩阵的求逆计算马尔可夫链的吸收时间。后一种策略以前在文献中提出过，但据我们所知，这是第一次用于显示预期运行时的非平凡边界。(2) 基本矩阵的求逆计算马尔可夫链的吸收时间。后一种策略以前在文献中提出过，但据我们所知，这是第一次用于显示预期运行时的非平凡边界。