First Mathematical Runtime Analyses of Multi-Objective Evolutionary Algorithms for Multi-Valued Decision Variables

2026-05-14

AI Summary

This work addresses the lack of rigorous runtime analysis for multi-objective evolutionary algorithms (MOEAs) with multi-valued (r > 2) decision variables. Focusing on the classical SEMO algorithm and its variants, we establish the first upper and lower bounds on the expected runtime required to converge to the Pareto front on the multi-valued OneMinMax problem. By integrating unit-strength local mutation with a strict dominance acceptance strategy and employing probabilistic and algorithmic theoretical techniques, we prove an upper bound of O(n²r² log n) for the standard SEMO. Furthermore, for an improved variant, we derive a tight bound of O(n²r(r + log n)), which matches the proven lower bound. This result fills a critical gap in the theoretical understanding of MOEAs operating in multi-valued search spaces.

Abstract

Problems defined on binary decision spaces have been intensively studied in the theory of multi-objective evolutionary algorithms (MOEAs). In contrast, no mathematical runtime analyses exist so far for MOEAs dealing with decision variables that take a finite number $r > 2$ of values, despite the prevalence of such problems in practice. In this work, we begin to fill this research gap. We analyze how the classic SEMO algorithm with unit-strength local mutation computes the Pareto front of an $r$-valued counterpart of the classic \oneminmax benchmark. For the expected number of function evaluations until the Pareto front is covered by the population of this MOEA, we prove an upper bound of $O(n^2 r^2 \log n)$ and a near-tight lower bound of $Ω(n^2 r (r + \log n))$. We can close the small remaining gap between these two bounds by considering a variant of the algorithm that accepts only strictly better solutions; for this variant, we show an upper bound of $O(n^2 r (r + \log n))$, matching our lower bound (which also holds for this variant). Our results suggest that classic MOEAs encounter no significant additional difficulties when dealing with multi-valued decision variables. However, significantly more advanced tools may be required to obtain tight bounds for algorithms with more complex population dynamics.

Problem

• multi-objective evolutionary algorithms
• multi-valued decision variables
• mathematical runtime analysis
• Pareto front
• combinatorial optimization

Innovation

• multi-objective evolutionary algorithms
• runtime analysis
• multi-valued decision variables
• Pareto front
• SEMO