Fast and Robust Consensus-Based Optimization via Optimal Feedback Control

SIAM Journal on Scientific Computing, vol. 48, pp. A74–A102

Abstract

Abstract. We propose a variant of consensus-based optimization (CBO) algorithms, controlled-CBO, which introduces a feedback control term to improve convergence towards global minimizers of nonconvex functions in multiple dimensions. The feedback law is a gradient of a numerical approximation to the Hamilton–Jacobi–Bellman (HJB) equation, which serves as a proxy of the original objective function. Thus, the associated control signal furnishes gradient-like information to facilitate the identification of the global minimum without requiring derivative computation from the objective function itself. The proposed method exhibits significantly improved performance over standard CBO methods in numerical experiments, particularly in scenarios involving a limited number of particles, or where the initial particle ensemble is not well positioned with respect to the global minimum. At the same time, the modification keeps the algorithm amenable to theoretical analysis in the mean-field sense. The superior convergence rates are assessed experimentally. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/AmberYuyangHuang/ControlledCBO and in the supplementary material ( M170644_Supplementary_Materials.pdf [259KB], ControlledCBO-main.zip [20.0KB]). [Formula: see text]

Authors 4

1. Yuyang Huang

   Imperial College London

   Affiliation as printed

   Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK

2. Michaël Herty Aachen Department of Mathematics and Applied Mathematics

   University of Pretoria · RWTH Aachen University

   Affiliation as printed

   IGPM, RWTH Aachen University, D-52062 Aachen, Germany, and Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, 0028, South Africa

3. Dante Kalise

   Imperial College London

   Affiliation as printed

   Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK

4. Nikolas Kantas

   Imperial College London

   Affiliation as printed

   Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ London, UK