The widely used multiobjective optimizer NSGA-II was recently proven to have considerable difficulties in many-objective optimization. In contrast, experimental results in the literature show a good performance of the SMS-EMOA, which can be seen as a steady-state NSGA-II that uses the hypervolume contribution instead of the crowding distance as the second selection criterion. This paper conducts the first rigorous runtime analysis of the SMS-EMOA for many-objective optimization. To this aim, we first propose a many-objective counterpart, the m-objective mOJZJ problem, of the bi-objective OJZJ benchmark, which is the first many-objective multimodal benchmark used in a mathematical runtime analysis. We prove that SMS-EMOA computes the full Pareto front of this benchmark in an expected number of $O(M^2 n^k)$ iterations, where $n$ denotes the problem size (length of the bit-string representation), $k$ the gap size (a difficulty parameter of the problem), and $M=(2n/m-2k+3)^{m/2}$ the size of the Pareto front. This result together with the existing negative result on 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, the stochastic population update often does not help for mOJZJ. It results in a $1/\Theta(\min\{Mk^{1/2}/2^{k/2},1\})$ speed-up, which is $\Theta(1)$ for large $m$ such as $m>k$. On the positive side, we prove that heavy-tailed mutation still results in a speed-up of order $k^{0.5+k-\beta}$. Finally, we conduct the first runtime analyses of the SMS-EMOA on the bi-objective OneMinMax and LOTZ benchmarks and show that it has a performance comparable to the GSEMO and the NSGA-II.

翻译：广泛使用的多目标优化器NSGA-II近期被证明在多目标优化中存在显著困难。相比之下，文献中的实验结果表明SMS-EMOA具有良好的性能——该算法可视为稳态NSGA-II的变体，其使用超体积贡献而非拥挤距离作为第二选择准则。本文首次对SMS-EMOA在多目标优化问题上的运行时进行严格分析。为此，我们首先提出双目标OJZJ基准测试的多目标对应版本——m目标mOJZJ问题，这是首个用于数学运行时分析的多目标多模态基准。我们证明，SMS-EMOA在期望$O(M^2 n^k)$次迭代内即可计算该基准的完整帕累托前沿，其中$n$表示问题规模（比特串表示长度），$k$为间隙大小（问题难度参数），$M=(2n/m-2k+3)^{m/2}$为帕累托前沿规模。该结果与关于原始NSGA-II的现有负面结论共同表明：原则上NSGA-II的通用框架适用于多目标优化，但其采用的拥挤距离作为平局决胜标准存在缺陷。我们进一步获得关于SMS-EMOA的三项新见解：不同于近期双目标OJZJ基准的研究结果，随机种群更新策略对mOJZJ问题通常并无助益——其带来的加速比为$1/\Theta(\min\{Mk^{1/2}/2^{k/2},1\})$，当$m>k$等大规模m情况下该值趋于$\Theta(1)$。在积极方面，我们证明重尾变异仍能产生$k^{0.5+k-\beta}$量级的加速效果。最后，我们首次对SMS-EMOA在双目标OneMinMax和LOTZ基准上的运行时进行分析，表明其性能与GSEMO和NSGA-II相当。