Regularity and positivity of solutions of the Consensus-Based Optimization equation: unconditional global convergence

TL;DR

This paper proves that solutions to the Consensus-Based Optimization (CBO) equation remain positive and fully supported, and establishes the unconditional global convergence of CBO methods to global minimizers, advancing theoretical understanding of this optimization technique.

Contribution

It provides the first proof of unconditional global convergence for CBO, based on regularity and positivity of solutions to the associated nonlinear Fokker-Planck equation.

Findings

Solutions to the CBO equation are positive and have full support.

Unconditional global convergence of CBO to global minimizers is established.

Existence of smooth solutions for a broader class of drift-diffusion equations is demonstrated.

Abstract

Introduced in 2017 \cite{B1-pinnau2017consensus}, Consensus-Based Optimization (CBO) has rapidly emerged as a significant breakthrough in global optimization. This straightforward yet powerful multi-particle, zero-order optimization method draws inspiration from Simulated Annealing and Particle Swarm Optimization. Using a quantitative mean-field approximation, CBO dynamics can be described by a nonlinear Fokker-Planck equation with degenerate diffusion, which does not follow a gradient flow structure. In this paper, we demonstrate that solutions to the CBO equation remain positive and maintain full support. Building on this foundation, we establish the {\it unconditional} global convergence of CBO methods to global minimizers. Our results are derived through an analysis of solution regularity and the proof of existence for smooth, classical solutions to a broader class of drift-diffusion equations, despite the challenges posed by degenerate diffusion.