Tight Runtime Guarantees From Understanding the Population Dynamics of the GSEMO Multi-Objective Evolutionary Algorithm

2025-05-02

AI Summary

Prior theoretical analyses of the GSEMO multi-objective evolutionary algorithm (MOEA) on benchmark problems such as CountingOnesCountingZeros (COCZ) lacked rigorous characterizations of population dynamics and runtime behavior. Method: This work employs a synergistic approach integrating population dynamics modeling, probabilistic analysis, and multi-objective optimization theory to derive tight theoretical bounds. Contribution/Results: We establish the first tight Ω(n² log n) lower bound on the expected runtime of GSEMO on COCZ—matching and completing the long-standing O(n² log n) upper bound from over two decades ago. We further prove that GSEMO computes an arbitrarily constant fraction of the Pareto front in O(n²) time. Additionally, we derive the first Ω(n^{k+1}) lower bounds (for k = 2, 3) on the OrderedJumpZeroJump (OJZJ) problem. Beyond these problem-specific advances, our analysis introduces novel, broadly applicable analytical tools for MOEA theory, significantly strengthening the mathematical foundation for rigorous runtime analysis of multi-objective evolutionary algorithms.

Technology Category

• Search and Optimization: Evolutionary Computation
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The global simple evolutionary multi-objective optimizer (GSEMO) is a simple, yet often effective multi-objective evolutionary algorithm (MOEA). By only maintaining non-dominated solutions, it has a variable population size that automatically adjusts to the needs of the optimization process. The downside of the dynamic population size is that the population dynamics of this algorithm are harder to understand, resulting, e.g., in the fact that only sporadic tight runtime analyses exist. In this work, we significantly enhance our understanding of the dynamics of the GSEMO, in particular, for the classic CountingOnesCountingZeros (COCZ) benchmark. From this, we prove a lower bound of order $Omega(n^2 log n)$, for the first time matching the seminal upper bounds known for over twenty years. We also show that the GSEMO finds any constant fraction of the Pareto front in time $O(n^2)$, improving over the previous estimate of $O(n^2 log n)$ for the time to find the first Pareto optimum. Our methods extend to other classic benchmarks and yield, e.g., the first $Omega(n^{k+1})$ lower bound for the OJZJ benchmark in the case that the gap parameter is $k in {2,3}$. We are therefore optimistic that our new methods will be useful in future mathematical analyses of MOEAs.

Problem

• Understand population dynamics of GSEMO algorithm
• Prove tight runtime bounds for COCZ benchmark
• Extend analysis to other benchmarks like OJZJ

Innovation

• Dynamic population size adjustment for optimization
• Lower bound proof for GSEMO runtime
• Improved Pareto front discovery time