On the mean-field limit for the consensus-based optimization

TL;DR

This paper establishes the mean-field limit for consensus-based optimization and particle swarm optimization by proving the convergence of empirical measures to a unique solution of the mean-field equations.

Contribution

It provides the first rigorous proof of the large particle limit for CBO and extends the results to PSO models.

Findings

Empirical measures are tight and converge to the mean-field solution.

The limit measure is shown to be the unique weak solution to the mean-field CBO equation.

Results confirm the validity of the mean-field approximation for large particle systems.

Abstract

This paper is concerned with the large particle limit for the consensus-based optimization (CBO), which was postulated in the pioneering works [6,28]. In order to solve this open problem, we adapt a compactness argument by first proving the tightness of the empirical measures $\{\mu^N\}_{N\geq 2}$ associated to the particle system and then verifying that the limit measure $\mu$ is the unique weak solution to the mean-field CBO equation. Such results are extended to the model of particle swarm optimization (PSO).