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，对多目标进化优化器NSGA-II的首批数学运行时分析已经完成。我们延续这一研究方向，首次分析了该算法在由两个多模态目标组成的基准问题上的运行时表现。我们证明，若种群规模$N$至少为帕累托前沿大小的四倍，则采用四种不同亲本选择方式及位变异操作的NSGA-II能够在$O(N n^k)$时间内优化跳跃规模为$2 \le k \le n/4$的OneJumpZeroJump基准问题。当使用快速变异（一种近期提出的重尾变异算子）时，该保证可进一步改进$k^{\Omega(k)}$倍。总体而言，本研究表明NSGA-II在处理OneJumpZeroJump问题的局部最优时，其性能至少与全局SEMO算法相当。