An Adaptive Consensus Based Method for Multi-objective Optimization with Uniform Pareto Front Approximation

Giacomo Borghi*, Michael Herty, Lorenzo Pareschi

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
28 Downloads (Pure)

Abstract

In this work we are interested in stochastic particle methods for multi-objective optimization. The problem is formulated via scalarization 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, in particular over the Pareto front, by using energy-based measures to quantify the diversity of the system. The resulting gradient-free metaheuristic algorithm is mathematically analyzed using a mean-field approximation of the algorithm iteration and convergence guarantees towards Pareto optimal points are rigorously proven. In addition, we analyze the dynamics when the Pareto front corresponds to the unit simplex, and show that the adaptive mechanism reduces to a gradient flow in this case. Several numerical experiments show the validity of the proposed stochastic particle dynamics, investigate the role of the algorithm parameters and validate the theoretical findings.

Original languageEnglish
Article number58
JournalApplied Mathematics and Optimization
Volume88
Issue number2
DOIs
Publication statusPublished - 10 Aug 2023

Keywords

  • Consensus-based optimization
  • Gradient-free methods
  • Mean-field limit
  • Multi-objective optimization
  • Stochastic particle methods

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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