SPEA2$^+$: Improved Density Estimation in SPEA2 with Provable Runtime Guarantees

2026-06-10

AI Summary

This work addresses the limited diversity in SPEA2 when handling dominating solutions, which stems from its reliance on k-nearest neighbor distances and hinders efficient coverage of complex Pareto fronts such as those in OneTrapZeroTrap problems. For the first time, the paper presents a rigorous runtime theoretical analysis of SPEA2’s mechanism for processing dominating solutions and introduces an enhanced algorithm, SPEA2⁺, which replaces k-nearest neighbor distances with all-pair distances to improve population diversity. Theoretical analysis demonstrates that SPEA2⁺ achieves superior runtime guarantees on complex problems. Empirical results confirm that SPEA2⁺ matches the performance of state-of-the-art algorithms on OneTrapZeroTrap while preserving SPEA2’s original strengths on simpler problems, with theory and experiments showing consistent agreement.

Abstract

The Strength Pareto Evolutionary Algorithm 2 (SPEA2) is a popular and prominent evolutionary algorithm for solving multi-objective optimisation problems. Despite its popularity, theoretical analyses of SPEA2 have only appeared recently. Moreover, these analyses focus exclusively on how SPEA2 handles non-dominated solutions and disregard the algorithmic components responsible for handling dominated solutions. We conduct a first runtime analysis of SPEA2 for which these components are analysed. We prove that, unlike other prominent algorithms, including NSGA-II, NSGA-III and SMS-EMOA under the same setting of constant population size and duplicate elimination, SPEA2 is unable to cover the Pareto front of the OneTrapZeroTrap benchmark efficiently. Our results indicate that using k-th nearest-neighbour distance in the fitness assignment provides an insufficient signal to maintain diversity among dominated individuals. To address this issue, we propose an improved variant, SPEA2$^+$, that considers all pairwise distances. The new algorithm achieves the same performance guarantees as the other prominent algorithms on OneTrapZeroTrap, while matching the performance of the original SPEA2 on simpler problems. Experimental results complement our theoretical findings.

Problem

• SPEA2
• multi-objective optimisation
• Pareto front
• diversity maintenance
• dominated solutions

Innovation

• SPEA2+
• density estimation
• runtime analysis
• multi-objective optimization
• pairwise distances