Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives
This work addresses a theoretical gap for researchers in evolutionary computation, providing foundational insights into MOEA performance on multimodal objectives, though it is incremental as it adapts existing single-objective techniques.
The paper tackles the lack of theoretical understanding for multiobjective evolutionary algorithms (MOEAs) on multimodal problems by proposing the OJZJ benchmark and analyzing algorithms like SEMO and GSEMO. It proves that GSEMO covers the Pareto front in expected Θ((n-2k)n^k) iterations and shows speed-ups of factor-5 to factor-10 with heavy-tailed mutation and stagnation detection for small problem sizes.
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 $Θ((n-2k)n^{k})$ 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 $k^{Ω(k)}$. 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 $k^{Ω(k)}$ 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.