AI 中文总结
研究多目标优化进化算法中动态种群规模作用，引入CLIMB问题类，分析GSEMO和NSGA-II运行时，证明NSGA-II动态种群规模有加速比，GSEMO和NSGA-II-DYN能更快找到帕累托前沿，是多目标优化中GSEMO超NSGA-II的首次严格分析。
AI 中文摘要
本文研究了动态种群规模在进化多目标优化中的作用。尽管此类方法在实践中广泛使用，但益处仍知之甚少，且缺乏严格的运行时分析。为此引入双目标问题类CLIMB，分析了GSEMO和NSGA-II在此问题上的运行时。结果表明，NSGA-II采用动态种群规模可适度改进，加速比为$\Omega(\sqrt{n}/\log n)$。特别地，证明了GSEMO和本文提出的带动态种群规模的NSGA-II-DYN能在期望$O(n \log n)$次适应度评估中找到CLIMB的帕累托前沿，而固定种群规模的NSGA-II期望需要$\Omega(n^{1.5})$次适应度评估。这是多目标优化中首次严格运行时分析证明GSEMO比NSGA-II有超常数加速比。分析基于单目标优化概念，采用家族三方法证明下界。
英文摘要
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.
