Very recently, the first mathematical runtime analyses of the multi-objective evolutionary optimizer NSGA-II have been conducted. We continue this line of research with a first runtime analysis of this algorithm on a benchmark problem consisting of two multimodal objectives. We prove that if the population size $N$ is at least four times the size of the Pareto front, then the NSGA-II with four different ways to select parents and bit-wise mutation optimizes the OneJumpZeroJump benchmark with jump size~$2 \le k \le n/4$ in time $O(N n^k)$. When using fast mutation, a recently proposed heavy-tailed mutation operator, this guarantee improves by a factor of $k^{\Omega(k)}$. Overall, this work shows that the NSGA-II copes with the local optima of the OneJumpZeroJump problem at least as well as the global SEMO algorithm.
翻译:最近,对多目标进化优化型NSGA-II进行了首次数学运行时间分析。我们继续这一研究线,对由两个多式联运目标组成的基准问题进行首次运行分析。我们证明,如果人口规模至少是Pareto前线规模的四倍,那么以四种不同方式选择父母和以比特方式进行突变的NSGA-II,则以跳跃大小~2克/列克/列恩/4美元的时间优化一个JumpZeroJump基准。当使用快速变异(最近提议的重尾突变操作者)时,这一保证通过一个单位的美元/奥姆加(k)美元来提高。总体而言,这项工作表明,NSGA-II至少要应对一个JumZeroJump问题的当地opima以及全球SEMO算法。</s>