Provable Speedups From Dynamic Population Sizes in Evolutionary Algorithms for Multiobjective Optimization

2026-07-26

AI Summary

This work addresses the lack of theoretical foundations for dynamic population sizing in multi-objective evolutionary algorithms by introducing a novel bi-objective benchmark problem, CLIMB. Through rigorous runtime analysis, it compares the performance of GSEMO and NSGA-II under both fixed and dynamic population strategies. The study provides the first provable super-constant speedup of GSEMO over fixed-population NSGA-II and proposes a new variant, NSGA-II-DYN. Leveraging diversity-based evolutionary analysis, family-tree lower-bound techniques, and tools from single-objective optimization theory, the paper establishes that both NSGA-II-DYN and GSEMO converge to the Pareto front in expected $O(n \log n)$ fitness evaluations, whereas fixed-population NSGA-II requires $\Omega(n^{1.5})$, yielding an asymptotic speedup of $\Omega(\sqrt{n} / \log n)$.

Abstract

This paper investigates the role of dynamic population sizes in evolutionary multi-objective optimization. Although such approaches are widely used in practice, their benefits remain poorly understood, and rigorous runtime analyses explaining when and why they help are still scarce. To address this, we introduce the bi-objective problem class CLIMB and analyze the runtime of GSEMO and the widely used NSGA-II on this problem. Our results show that allowing a dynamic population size for NSGA-II can lead to a moderate improvement, yielding a speedup of order $Ω(\sqrt{n}/\log n)$. In particular, we prove that GSEMO and NSGA-II-DYN, a version of NSGA-II with dynamic population sizes we propose in this paper, can find the Pareto front of CLIMB in expected $O(n \log n)$ fitness evaluations, whereas NSGA-II with a fixed population size requires $Ω(n^{1.5})$ fitness evaluations in expectation. To the best of our knowledge, this is the first rigorous runtime analysis in multi-objective optimization demonstrating a super-constant speedup of GSEMO over NSGA-II. Our analysis builds on concepts from single-objective optimization, like the evolution of population diversity over time, and employs the well-known family-three method to prove the lower bound.

Problem

• dynamic population sizes
• evolutionary multi-objective optimization
• runtime analysis
• NSGA-II
• GSEMO

Innovation

• dynamic population size
• multi-objective optimization
• runtime analysis
• NSGA-II
• GSEMO