Multiobjective evolutionary algorithms are successfully applied in many real-world multiobjective optimization problems. As for many other AI methods, the theoretical understanding of these algorithms 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 $OneJumpZeroJump$ problem, a bi-objective problem composed of two objectives isomorphic to the classic jump function benchmark. We prove that the simple evolutionary multiobjective optimizer (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 $k\in [4..\frac{n}{2}-1]$⁠, the global SEMO (GSEMO) covers the Pareto front in an expected number of $\Theta \left((n-2k)n^k\right)$ iterations. For $k=o\left(n\right)$⁠, we also show the tighter bound $\frac{3}{2}e{n}^{k+1}\pm o\left(n^{k+1}\right)$⁠, 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 $k^{\Omega \left(k\right)}$⁠. 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 $k^{\Omega \left(k\right)}$ 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.

多目标进化算法已成功应用于许多现实世界的多目标优化问题。至于许多其他人工智能方法，对这些算法的理论理解远远落后于它们在实践中的成功。特别是，以前的理论工作主要考虑由单峰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$氧neJ你米pZer哦J你米p$问题，由与经典跳跃函数基准同构的两个目标组成的双目标问题。我们证明，无论运行时间如何，概率为 1 的简单进化多目标优化器 (SEMO) 都不会计算完整的帕累托前沿。相反，对于所有问题规模$n$以及所有跳跃尺寸$k\epsilon [4。。\frac{n}{2}-1]$⁠，全局 SEMO (GSEMO) 以预期数量覆盖帕累托前沿$\theta （（n-2k）n^k）$迭代。为了$k=哦（n）$⁠，我们还展示了更紧的界限$\frac{3}{2}e{n}^{k+1}\pm 哦（n^{k+1}）$⁠，这可能是与低阶项紧密分开的 MOEA 的第一个运行时约束。我们还将 GSEMO 与两种在单目标多模态问题中显示出优势的方法相结合。当将 GSEMO 与重尾变异算子结合使用时，预期运行时间至少提高了 1 倍$k^{\Omega （k）}$⁠ . 当将 Rajabi 和 Witt (2022) 最近的停滞检测策略应用于 GSEMO 时，预期运行时间也至少提高了 1 倍$k^{\Omega （k）}$并以一个小的多项式因子超越了重尾 GSEMO$k$⁠ . 通过实验分析，我们表明，对于小问题规模，这些渐近差异已经可见：对于跳跃规模 4，已经可以观察到重尾突变的 5 倍加速和停滞检测的 10 倍加速问题规模在 10 到 50 之间。总的来说，我们的结果表明，最近开发的用于帮助单目标进化算法处理局部最优的想法也可以有效地应用于多目标优化。