The classic NSGA-II was recently proven to have considerable difficulties in many-objective optimization. This paper conducts the first rigorous runtime analysis in many objectives for the SMS-EMOA, a steady-state NSGA-II that uses the hypervolume contribution instead of the crowding distance as the second selection criterion. To this aim, we first propose a many-objective counterpart, the m-objective mOJZJ, of the bi-objective OJZJ, which is the first many-objective multimodal benchmark for runtime analysis. We prove that SMS-EMOA computes the full Pareto front of this benchmark in an expected number of $O(\mu M n^k)$ iterations, where $n$ denotes the problem size (length of the bit-string representation), $k$ the gap size (a difficulty parameter of the problem), $M=(2n/m-2k+3)^{m/2}$ the size of the Pareto front, and $\mu$ the population size (at least the same size as the largest incomparable set). This result together with the existing negative result for the original NSGA-II shows that, in principle, the general approach of the NSGA-II is suitable for many-objective optimization, but the crowding distance as tie-breaker has deficiencies. We obtain three additional insights on the SMS-EMOA. Different from a recent result for the bi-objective OJZJ benchmark, a recently proposed stochastic population update often does not help for mOJZJ. It at most results in a speed-up by a factor of order $2^{k} / \mu$, which is $\Theta(1)$ for large $m$, such as $m>k$. On the positive side, we prove that heavy-tailed mutation irrespective of the number $m$ of objectives results in a speed-up of order $k^{0.5+k-\beta}/e^k$, the same advantage as previously shown for the bi-objective case. Finally, we conduct the first runtime analyses of the SMS-EMOA on the classic bi-objective OneMinMax and LOTZ benchmarks and show that the SMS-EMOA has a performance comparable to the GSEMO and the NSGA-II.

翻译：经典的NSGA-II近期被证明在多目标优化中存在显著困难。本文首次对SMS-EMOA——一种采用超体积贡献度替代拥挤距离作为第二选择准则的稳态NSGA-II——进行了多目标场景下的严格运行时分析。为此，我们首先提出了双目标OJZJ基准问题的多目标版本，即m目标mOJZJ，这是首个用于运行时分析的多目标多模态基准问题。我们证明了SMS-EMOA在期望迭代次数$O(\mu M n^k)$内可计算该基准问题的完整帕累托前沿，其中$n$表示问题规模（比特串表示的长度），$k$为间隙大小（问题的难度参数），$M=(2n/m-2k+3)^{m/2}$为帕累托前沿的规模，$\mu$为种群规模（至少需达到最大不可比集的规模）。该结果与现有关于原始NSGA-II的负面结论共同表明：NSGA-II的基本框架原则上适用于多目标优化，但其采用的拥挤距离作为平局决胜机制存在缺陷。我们进一步获得了关于SMS-EMOA的三项洞见：首先，与近期双目标OJZJ基准的研究结果不同，新近提出的随机种群更新策略对mOJZJ通常并无助益，其至多能带来$2^{k} / \mu$量级的加速，而对于较大$m$（例如$m>k$）该加速效果仅为$\Theta(1)$。其次，我们证明了无论目标数$m$如何，重尾变异均能带来$k^{0.5+k-\beta}/e^k$量级的加速，这与先前在双目标场景中证实的优势一致。最后，我们首次在经典双目标基准问题OneMinMax和LOTZ上对SMS-EMOA进行了运行时分析，结果表明SMS-EMOA具有与GSEMO和NSGA-II相当的性能。