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

Applied Mathematics & Optimization, vol. 88

Abstract

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.

Authors 3

1. Giacomo Borghi corresponding Aachen Institute for Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Practical Mathematics, RWTH Aachen University, Aachen, Germany

2. Michaël Herty Aachen Institute for Geometry and Practical Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Practical Mathematics, RWTH Aachen University, Aachen, Germany

3. Lorenzo Pareschi

   University of Ferrara

   Affiliation as printed

   Department of Mathematics and Computer Science & Center for Modelling, Computing and Statistics, University of Ferrara, Ferrara, Italy