The theoretical understanding of MOEAs is lagging far behind their success in practice. In particular, previous theory work considers mostly easy problems that are composed of unimodal objectives. As a first step towards a deeper understanding of how evolutionary algorithms solve multimodal multiobjective problems, we propose the OJZJ problem, a bi-objective problem composed of two objectives isomorphic to the classic jump function benchmark. We prove that SEMO with probability one does not compute the full Pareto front, regardless of the runtime. In contrast, for all problem sizes $n$ and all jump sizes ${k \in [4..\frac n2 - 1]}$, the global SEMO (GSEMO) covers the Pareto front in an expected number of $\Theta((n-2k)n^{k})$ iterations. For $k = o(n)$, we also show the tighter bound $\frac 32 e n^{k+1} \pm o(n^{k+1})$, which might be the first runtime bound for an MOEA that is tight apart from lower-order terms. We also combine the GSEMO with two approaches that showed advantages in single-objective multimodal problems. When using the GSEMO with a heavy-tailed mutation operator, the expected runtime improves by a factor of at least $k^{\Omega(k)}$. When adapting the recent stagnation-detection strategy of Rajabi and Witt (2022) to the GSEMO, the expected runtime also improves by a factor of at least $k^{\Omega(k)}$ and surpasses the heavy-tailed GSEMO by a small polynomial factor in $k$. Via an experimental analysis, we show that these asymptotic differences are visible already for small problem sizes: A factor-$5$ speed-up from heavy-tailed mutation and a factor-$10$ speed-up from stagnation detection can be observed already for jump size~$4$ and problem sizes between $10$ and $50$. Overall, our results show that the ideas recently developed to aid single-objective evolutionary algorithms to cope with local optima can be effectively employed also in multiobjective optimization.

翻译：多目标进化算法（MOEAs）的理论理解仍远落后于其在实践中的成功。特别是，以往的理论研究主要集中在由单模态目标组成的简单问题上。作为深入理解进化算法如何求解多模态多目标问题的第一步，我们提出了OJZJ问题，这是一个由两个与经典跳跃函数基准同构的目标构成的双目标问题。我们证明，无论运行多长，SEMO算法以概率1无法计算出完整的帕累托前沿。相反，对于所有问题规模$n$和所有跳跃规模${k \in [4..\frac n2 - 1]}$，全局SEMO（GSEMO）能够在期望的$\Theta((n-2k)n^{k})$次迭代内覆盖帕累托前沿。对于$k = o(n)$，我们还给出了更紧的界$\frac 32 e n^{k+1} \pm o(n^{k+1})$，这可能是首个除了低阶项外紧致成立的多目标进化算法运行时界。我们还将GSEMO与两种在单目标多模态问题中表现出优势的方法相结合。当使用带重尾变异算子的GSEMO时，期望运行时间至少改善了$k^{\Omega(k)}$倍。当将Rajabi与Witt（2022）的最新停滞检测策略适配到GSEMO时，期望运行时间也至少改善了$k^{\Omega(k)}$倍，并且其性能略优于重尾GSEMO $k$的一个小多项式因子。通过实验分析，我们证明这些渐近差异在较小问题规模上已可见：对于跳跃规模~$4$以及规模在$10$到$50$之间的问题，重尾变异可带来~5倍的加速，而停滞检测可带来~10倍的加速。总体而言，我们的结果表明，近期为帮助单目标进化算法应对局部最优而提出的思想，同样可以在多目标优化中得到有效应用。