关于跳跃函数上$(1+(λ,λ))$遗传算法从局部最优逃逸时间的渐近分析
Asymptotical Analysis of the $(1+(λ,λ))$ GA Escape Time from Local Optima on Jump Functions
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中文总结 AI 辅助
研究基于概率论极限定理分析进化算法运行时间，针对Jump\(_k\)基准函数族，考虑\((1+(\lambda,\lambda))GA\)算法，在\(np\)趋于无穷时研究其从高原逃逸时间，收紧了逃逸时间上界并扩大了参数适用范围。
中文摘要 AI 辅助
本文基于概率论的极限定理，开发了一种进化算法运行时间分析方法。我们考虑在长度为\(n\)的二进制字符串搜索空间上定义的Jump\(_k\)基准函数族，由整数\(k\)参数化，在与唯一全局最优解的汉明距离\(k\)处有多个局部最优解。本文考虑了具有可调突变率\(p\)、交叉偏差\(c\)以及两个中间种群大小\(\lambda_M\)和\(\lambda_C\)的遗传算法\((1+(\lambda,\lambda))GA\)，研究了在\(np\)趋于无穷大时，在Jump\(_k\)适应度函数情况下从高原逃逸的时间。本文的主要结果是收紧了Antipov、Doerr和Karavaev（2022）工作中逃逸时间的上界。此外，得到的界适用于更广泛的算法参数范围。
英文摘要
The paper develops the approach to the runtime analysis of evolutionary algorithms on the basis of limit theorems from probability theory. We consider the family of Jump$_k$ benchmark functions, defined on the search space of binary strings of length $n$, parametrized by the integer $k$, which have a plateau of multiple local optima at the Hamming distance $k$ from a unique global optimum. In this work, we consider the genetic algorithm $(1+(λ,λ)) GA$ from (Doerr, Doerr and Ebel, 2015) with tunable parameters of the mutation rate $p$, crossover bias $c$, and two intermediate population sizes $λ_M$ and $λ_C$. We study the time it escapes from the plateau of local optima and reaches the global optimum in the case of Jump$_k$ fitness function and tighten the upper bounds on the expected escape time, known from the work of Antipov, Doerr and Karavaev (2022). The obtained bounds also apply to a wider range of algorithmic parameters. The main result of this work applies to the case when $k\to \infty$ as $n \to \infty.$ The case of finite $k$ is investigated quite simply and considered tangentially.
