多约束黑箱模型中的OneMax
OneMax in Black-Box Models with Several Restrictions
摘要：
黑箱复杂性研究通用黑箱优化算法（如进化算法和其他搜索算法）的效率下限。存在不同的模型，每个模型都被设计用于分析典型启发式方法的不同方面，例如内存大小或使用的变异运算符。虽然大多数以前的作品集中在一个特定的这样的方面，我们认为在这项工作中如何结合几个算法的限制影响黑盒的复杂性。我们的测试平台是所谓的OneMax函数，这是一组经典的测试函数，与经典的硬币称重问题和棋盘游戏Mastermind密切相关。我们特别分析了OneMax在不同内存大小下的组合内存受限的基于排名的黑盒复杂度。虽然其隔离的存储器限制以及其基于排名的黑盒复杂度为长度$n$的位串仅为$n/\log n$阶，但组合模型不允许算法比$n$中的线性算法更快，如标准信息论考虑所示。我们证明了这个线性界确实是渐近紧的。对于其他内存和子内存大小，也得到了类似的结果。我们的结果也适用于最近引入的精英模型中OneMax的（Monte Carlo）复杂性，其中只有迄今为止最好的解决方案可以保存在内存中。最后，我们还提供了改进的下界的OneMax在所考虑的模型的复杂性。我们的结果活跃了自然进化算法优化OneMax在$o(n \log n)$迭代的追求。
英文摘要：
Black-box complexity studies lower bounds for the efficiency of general-purpose black-box optimization algorithms such as evolutionary algorithms and other search heuristics. Different models exist, each one being designed to analyze a different aspect of typical heuristics such as the memory size or the variation operators in use. While most of the previous works focus on one particular such aspect, we consider in this work how the combination of several algorithmic restrictions influence the black-box complexity. Our testbed are so-called OneMax functions, a classical set of test functions that is intimately related to classic coin-weighing problems and to the board game Mastermind. We analyze in particular the combined memory-restricted ranking-based black-box complexity of OneMax for different memory sizes. While its isolated memory-restricted as well as its ranking-based black-box complexity for bit strings of length $n$ is only of order $n/\log n$, the combined model does not allow for algorithms being faster than linear in $n$, as can be seen by standard information-theoretic considerations. We show that this linear bound is indeed asymptotically tight. Similar results are obtained for other memory- and offspring-sizes. Our results also apply to the (Monte Carlo) complexity of OneMax in the recently introduced elitist model, in which only the best-so-far solution can be kept in the memory. Finally, we also provide improved lower bounds for the complexity of OneMax in the regarded models. Our result enlivens the quest for natural evolutionary algorithms optimizing OneMax in $o(n \log n)$ iterations.
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