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：近期，针对多目标进化优化器NSGA-II的数学运行时间分析首次得以开展。我们延续这一研究方向，对包含两个多模态目标的基准问题进行首次算法运行时间分析。我们证明：若种群大小$N$至少为帕累托前沿大小的四倍，则采用四种不同父代选择策略及逐位变异的NSGA-II算法，可在$O(N n^k)$时间内优化跳跃规模$2 \le k \le n/4$的OneJumpZeroJump基准问题。当采用快速变异（一种近期提出的重尾变异算子）时，该保证可提升$k^{\Omega(k)}$倍。总体而言，本研究揭示了NSGA-II在应对OneJumpZeroJump问题的局部最优时，其性能至少与全局SEMO算法相当。