Regularity and positivity of solutions of the Consensus-Based Optimization equation: unconditional global convergence
Massimo Fornasier, Lukang Sun
TL;DR
The paper studies Consensus-Based Optimization (CBO), a zero-order multi-particle global optimization method, by analyzing its mean-field limit as a nonlinear Fokker–Planck equation with degenerate diffusion. It develops a general drift–diffusion framework, proves existence, uniqueness, and regularity of classical and weak solutions via Galerkin approximations, and then applies these results to the CBO dynamics, obtaining unconditional global convergence to global minimizers. A key contribution is establishing positivity and full support of the solution density for dimensions $d>1$, which removes previous technical requirements on the initial mass near minimizers. The findings provide rigorous theoretical guarantees for CBO in broad settings, including extensions to manifolds, and offer a foundational understanding of the algorithm’s convergence behavior with degenerate diffusion.
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.