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：多目标进化算法的理论理解远远落后于它们在实践中的成功。特别是，先前的理论研究大多考虑由单模态目标组成的简单问题。作为深入理解进化算法如何解决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组成的双目标问题。我们证明，无论运行时间如何，SEMO算法以概率1无法计算出完整的帕累托前沿。相比之下，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）算法在期望的$\Theta((n-2k)n^{k})$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$\frac 32 e n^{k+1} \pm o(n^{k+1})$，这可能是多目标进化算法中首个除低阶项外紧的运行时界。我们还结合了两种在单目标多模态问题中表现出优势的方法来改进GSEMO。当使用带重尾变异算子的GSEMO时，期望运行时间改善了至少$k^{\Omega(k)}$倍。当将Rajabi和Witt（2022）提出的停滞检测策略应用于GSEMO时，期望运行时间也改善了至少$k^{\Omega(k)}$倍，并且比重尾GSEMO在$k$上多出一个小多项式因子。通过实验分析，我们表明这些渐近差异在小问题规模下已经可见：对于跳跃规模$4$和问题规模在$10$到$50$之间，可以观察到重尾变异带来的$5$倍加速和停滞检测带来的$10$倍加速。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想法也可以有效应用于多目标优化。