Theoretical Analyses of Multiobjective Evolutionary Algorithms on Multimodal Objectives

TL;DR

This paper introduces OneJumpZeroJump as a bi-objective multimodal benchmark to study MOEAs and shows that SEMO cannot achieve the full Pareto front, while GSEMO can with an expected runtime of $O((n-2k)n^{k})$, and for $k=o(n)$, the bound tightens to $\frac{3}{2}e n^{k+1}$. It reveals that two single-objective ideas for multimodality—heavy-tailed mutation and stagnation-detection—transfer effectively to MOEAs, yielding superpolynomial improvements in special regimes: $k^{\Omega(k)}$ factors over standard GSEMO. The paper provides rigorous runtime analyses for GSEMO, GSEMO-HTM, and the SD-GSEMO on OneJumpZeroJump, plus nontrivial non-asymptotic constants, and corroborates the theory with experiments showing practical speedups of roughly 5×–10× for moderate problem sizes. Overall, the work demonstrates that recent multimodality techniques from single-objective optimization can substantially enhance multiobjective MOEAs, and paves the way for analyzing more advanced MOEAs and benchmarks.

Abstract

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.

Figures (4)

• Figure 1: The values of $f_1$ and $f_2$ with respect to $|x|_1$, the number of ones in the search point $x$.
• Figure 2: The values of two objectives in $\textsc{OneJumpZeroJump}\xspace_{n,k}$ with respect to $|x|_1$, the number of ones in the search point $x$.
• Figure 3: The Pareto front for the $\textsc{OneJumpZeroJump}\xspace_{n,k}$ function with $(n,k)=(50,10)$.
• Figure 4: The median number of function evaluations (with the first and third quartiles) of GSEMO, GSEMO-HTM, SD-GSEMO, and SD-GSEMO-Ind on $\textsc{OneJumpZeroJump}\xspace_{n,k}$ with $k=4$ and $n=10:4:50$ in 20 independent runs.

Theorems & Definitions (42)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 3: ZPLG QianTZ16
• Definition 4: SPG QianTZ16
• Definition 5: Dec-obj-MOP LiZZZ16
• Definition 6: $\textsc{OneJumpZeroJump}\xspace_{n,k}$
• Theorem 7
• proof