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arXiv 2203.11550cs.NE
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采用二进制和随机竞赛选择的非支配排序遗传算法II(NSGA-II)的运行时间分析

Running Time Analysis of the Non-dominated Sorting Genetic Algorithm II (NSGA-II) using Binary or Stochastic Tournament Selection

Chao Bian, Chao Qian

摘要:

进化算法已被广泛应用于解决多目标优化问题,并已成为最流行的工具。然而,多目标进化分析的理论基础,特别是其基本理论方面,运行时间分析,在很大程度上仍然不发达。现有的几个理论工作主要考虑简单的MOEA,而非支配排序遗传算法II(NSGA-II),可能是最有影响力的MOEA,还没有分析,除了最近的工作考虑一个简化的变体没有交叉。在本文中,我们提出了一个运行时间分析的标准NSGA-II解决LOTZ,OneMinMax和COCZ,三个常用的双目标优化问题。具体来说,我们证明了预期的运行时间(即,适合度评估的数量)是LOTZ的$O(n^3)$,以及OneMinMax和COCZ的$O(n^2\log n)$,这令人惊讶地与先前分析的简单MOEA、GSEMO和SEMO的相同。接下来,我们介绍一种新的父选择策略,随机锦标赛选择(即,$k$锦标赛选择,其中$k$是均匀随机抽样),以取代NSGA-II的二进制锦标赛选择策略,减少所需的预期运行时间为$O(n^2)$的所有三个问题。实验结果表明,LOTZ的运行时间上界是紧的,OneMinMax和COCZ的运行时间上界几乎是紧的.

英文摘要:

Evolutionary algorithms (EAs) have been widely used to solve multi-objective optimization problems, and have become the most popular tool. However, the theoretical foundation of multi-objective EAs (MOEAs), especially the essential theoretical aspect, i.e., running time analysis, has been still largely underdeveloped. The few existing theoretical works mainly considered simple MOEAs, while the non-dominated sorting genetic algorithm II (NSGA-II), probably the most influential MOEA, has not been analyzed except for a very recent work considering a simplified variant without crossover. In this paper, we present a running time analysis of the standard NSGA-II for solving LOTZ, OneMinMax and COCZ, the three commonly used bi-objective optimization problems. Specifically, we prove that the expected running time (i.e., number of fitness evaluations) is $O(n^3)$ for LOTZ, and $O(n^2\log n)$ for OneMinMax and COCZ, which is surprisingly as same as that of the previously analyzed simple MOEAs, GSEMO and SEMO. Next, we introduce a new parent selection strategy, stochastic tournament selection (i.e., $k$ tournament selection where $k$ is uniformly sampled at random), to replace the binary tournament selection strategy of NSGA-II, decreasing the required expected running time to $O(n^2)$ for all the three problems. Experiments are also conducted, suggesting that the derived running time upper bounds are tight for LOTZ, and almost tight for OneMinMax and COCZ.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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