An adaptive consensus based method for multi-objective optimization with uniform Pareto front approximation

TL;DR

This paper introduces an adaptive consensus-based stochastic particle method for multi-objective optimization that ensures uniform Pareto front approximation, with theoretical guarantees and numerical validation.

Contribution

It proposes a novel adaptive consensus-based algorithm with energy-based diversity measures and provides rigorous convergence analysis and gradient flow insights.

Findings

Algorithm achieves uniform Pareto front coverage.

Theoretical convergence guarantees are established.

Numerical experiments validate the method's effectiveness.

Abstract

In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated using parametrized, single-objective sub-problems which are solved simultaneously. To this end a consensus based multi-objective optimization method on the search space combined with an additional heuristic strategy to adapt parameters during the computations is proposed. The adaptive strategy aims to distribute the particles uniformly over the image space by using energy-based measures to quantify the diversity of the system. The resulting metaheuristic algorithm is mathematically analyzed using a mean-field approximation and convergence guarantees towards optimal points is rigorously proven. In addition, a gradient flow structure in the parameter space for the adaptive method is revealed and analyzed. Several numerical experiments shows the validity of the proposed stochastic particle dynamics and illustrate the theoretical findings.