Regularity and Positivity of Solutions of the Consensus-Based Optimization Equation: Unconditional Global Convergence

MCML Authors

Massimo Fornasier

Prof. Dr.

Principal Investigator

Applied Numerical Analysis

Lukang Sun

→ Group Massimo Fornasier
 Applied Numerical Analysis

Abstract

Introduced in 2017, 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 { 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.

misc

---

Preprint

Feb. 2025

---

Authors

M. Fornasier • L. Sun

---

Links

---

Research Area

 A2 | Mathematical Foundations