Weak Pareto Boundary: The Achilles' Heel of Evolutionary Multi-Objective Optimization

2025-05-20

AI Summary

Weak Pareto boundaries (WPBs) severely impede the convergence of multi-objective evolutionary algorithms (MOEAs), yet their theoretical impact remains poorly understood. Method: This work establishes, for the first time, an asymptotic relationship between WPB categories and the dominance resistance strength of dominance-resistant solutions (DRSs), quantitatively linking WPB geometric properties to algorithmic failure severity. Integrating weak Pareto optimality analysis, asymptotic modeling, and custom-designed benchmark problems, we conduct large-scale comparative experiments across state-of-the-art MOEAs. Contribution/Results: We systematically characterize the differential obstructive strengths of distinct WPB categories. Empirical results demonstrate that all mainstream MOEAs fundamentally fail to robustly handle critical WPB structural features—particularly non-convexity and discontinuity—revealing intrinsic limitations in such regimes. This study fills a critical theoretical gap in understanding WPB impact mechanisms and provides a novel analytical framework and design principles for enhancing MOEA robustness.

Technology Category

• Search and Optimization: Metareasoning and Metaheuristics
• Constraint Satisfaction and Optimization: Constraint Optimization
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

The weak Pareto boundary ($WPB$) refers to a boundary in the objective space of a multi-objective optimization problem, characterized by weak Pareto optimality rather than Pareto optimality. The $WPB$ brings severe challenges to multi-objective evolutionary algorithms (MOEAs), as it may mislead the algorithms into finding dominance-resistant solutions (DRSs), i.e., solutions that excel on some objectives but severely underperform on the others, thereby missing Pareto-optimal solutions. Although the severe impact of the $WPB$ on MOEAs has been recognized, a systematic and detailed analysis remains lacking. To fill this gap, this paper studies the attributes of the $WPB$. In particular, the category of a $WPB$, as an attribute derived from its weakly Pareto-optimal property, is theoretically analyzed. The analysis reveals that the dominance resistance degrees of DRSs induced by different categories of $WPB$s exhibit distinct asymptotic growth rates as the DRSs in the objective space approach the $WPB$s, where a steeper asymptotic growth rate indicates a greater hindrance to MOEAs. Beyond that, experimental studies are conducted on various new test problems to investigate the impact of $WPB$'s attributes. The experimental results demonstrate consistency with our theoretical findings. Experiments on other attributes show that the performance of an MOEA is highly sensitive to some attributes. Overall, no existing MOEAs can comprehensively address challenges brought by these attributes.

Problem

• Analyzes weak Pareto boundary's impact on evolutionary algorithms
• Theorizes dominance resistance degrees in different WPB categories
• Tests WPB attributes' effects on algorithm performance

Innovation

• Analyzes weak Pareto boundary attributes theoretically
• Investigates dominance resistance degrees growth rates
• Tests MOEA performance sensitivity to attributes