Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives*

2020-12-14

Evolutionary Computation

AI Summary

This work addresses the theoretical lag of multi-objective evolutionary algorithms (MOEAs) in multimodal multi-objective optimization. We introduce the OneJumpZeroJump bi-objective benchmark— the first theoretically grounded multimodal multi-objective problem—and analyze the GSEMO algorithm on it using drift analysis, heavy-tailed mutation, and stagnation detection. We establish the first tight asymptotic runtime bound for MOEAs: GSEMO covers the Pareto front in Θ((n−2k)nᵏ) expected iterations. Heavy-tailed mutation and stagnation detection each yield k^Ω(k)-factor speedups. Empirical validation on small-scale instances confirms up to 5× and 10× acceleration, respectively. Our contributions are threefold: (i) proposing the first multimodal multi-objective theoretical benchmark; (ii) developing the first tight runtime analysis framework for multi-objective optimization; and (iii) successfully transferring and rigorously validating advanced single-objective techniques—namely, heavy-tailed mutation and stagnation detection—in the multi-objective setting.

Abstract

Abstract 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∈[4..n2-1], the global SEMO (GSEMO) covers the Pareto front in an expected number of Θ((n-2k)nk) iterations. For k=o(n), we also show the tighter bound 32enk+1±o(nk+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 kΩ(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 kΩ(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.

Problem

• Analyze MOEAs on multimodal objectives
• Propose OJZJ problem for deeper understanding
• Improve GSEMO runtime with new strategies

Innovation

• GSEMO optimizes multimodal multiobjective problems
• Heavy-tailed mutation enhances GSEMO performance
• Stagnation detection improves GSEMO runtime