A consensus-based model for global optimization and its mean-field limit

TL;DR

This paper introduces a new consensus-based stochastic swarm intelligence model for global optimization, analyzing its mean-field limit through PDE methods and demonstrating its practical effectiveness via numerical experiments.

Contribution

It presents a novel consensus-based optimization algorithm with a rigorous mean-field limit analysis and convergence results, advancing the understanding of swarm intelligence methods for global optimization.

Findings

Convergence of the SI model to optimal solutions.

Derivation of a nonlocal PDE as the mean-field limit.

Numerical evidence supporting the algorithm's effectiveness.

Abstract

We introduce a novel first-order stochastic swarm intelligence (SI) model in the spirit of consensus formation models, namely a consensus-based optimization (CBO) algorithm, which may be used for the global optimization of a function in multiple dimensions. The CBO algorithm allows for passage to the mean-field limit, which results in a nonstandard, nonlocal, degenerate parabolic partial differential equation (PDE). Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the SI model. We further present numerical investigations underlining the feasibility of our approach.