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

TL;DR

This paper provides the first mathematical runtime analysis of multi-objective evolutionary algorithms for problems with multi-valued decision variables, extending understanding beyond binary decision spaces.

Contribution

It analyzes the SEMO algorithm's efficiency on multi-valued decision variables, establishing tight bounds on the expected runtime to cover the Pareto front.

Findings

Expected runtime bounds are $O(n^2 r^2 \log n)$ and $\Omega(n^2 r (r + \log n))$ for the classic SEMO algorithm.

A variant accepting only strictly better solutions achieves a tight bound of $O(n^2 r (r + \\log n))$.

Classic MOEAs face no significant additional difficulties with multi-valued decision variables.

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 $\Omega(n^2 r (r + \log n))$. 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.