Data-Driven Analysis of Pareto Set Topology

TL;DR

This paper introduces a data-driven method to analyze the topology of Pareto sets in multi-objective optimization, enabling the recognition of whether the Pareto set forms a topological simplex based on population data.

Contribution

It provides a novel theory and algorithm for identifying Pareto set topology from data, aiding in understanding EMO algorithm coverage capabilities.

Findings

Accurately recognizes Pareto set topology in high-dimensional problems

Works with reasonable population sizes

Validates method through numerical experiments

Abstract

When and why can evolutionary multi-objective optimization (EMO) algorithms cover the entire Pareto set? That is a major concern for EMO researchers and practitioners. A recent theoretical study revealed that (roughly speaking) if the Pareto set forms a topological simplex (a curved line, a curved triangle, a curved tetrahedron, etc.), then decomposition-based EMO algorithms can cover the entire Pareto set. Usually, we cannot know the true Pareto set and have to estimate its topology by using the population of EMO algorithms during or after the runtime. This paper presents a data-driven approach to analyze the topology of the Pareto set. We give a theory of how to recognize the topology of the Pareto set from data and implement an algorithm to judge whether the true Pareto set may form a topological simplex or not. Numerical experiments show that the proposed method correctly recognizes the topology of high-dimensional Pareto sets within reasonable population size.