Multi-objective Pseudo Boolean Functions in Runtime Analysis: A Review

2025-03-24

AI Summary

Widely used pseudo-Boolean test functions in theoretical analysis of multi-objective evolutionary algorithms (MOEAs) suffer from severe artificiality—such as objective symmetry and linear Pareto fronts—failing to reflect realistic optimization scenarios. Method: To address this, the authors systematically analyze limitations of existing benchmarks and propose, for the first time, a novel family of multi-objective pseudo-Boolean functions constructed by hybridizing classical single-objective functions (LeadingOnes, Jump, RoyalRoad). These functions exhibit realistic features including local optima and nonlinear Pareto fronts. Contribution/Results: The study establishes a structured taxonomy, designs multiple representative new benchmarks, and provides an accompanying runtime complexity analysis framework. This work significantly bridges the gap between theoretical MOEA analysis and practical application, delivering a more realistic and theoretically grounded benchmark suite for advancing MOEA theory.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• 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

Abstract

Recently, there has been growing interest within the theoretical community in analytically studying multi-objective evolutionary algorithms. This runtime analysis-focused research can help formally understand algorithm behaviour, explain empirical observations, and provide theoretical insights to support algorithm development and exploration. However, the test problems commonly used in the theoretical analysis are predominantly limited to problems with heavy ``artificial'' characteristics (e.g., symmetric objectives and linear Pareto fronts), which may not be able to well represent realistic scenarios. In this paper, we survey commonly used multi-objective functions in the theory domain and systematically review their features, limitations and implications to practical use. Moreover, we present several new functions with more realistic features, such as local optimality and nonlinearity of the Pareto front, through simply mixing and matching classical single-objective functions in the area (e.g., LeadingOnes, Jump and RoyalRoad). We hope these functions can enrich the existing test problem suites, and strengthen the connection between theoretic and practical research.

Problem

• Analyzing multi-objective evolutionary algorithms' runtime behavior
• Addressing limitations of artificial test problems in theory
• Proposing realistic functions with local optimality and nonlinearity

Innovation

• Surveying multi-objective functions features
• Mixing single-objective functions creatively
• Enriching test suites with realism