A PDE Theory for Consensus-Based Optimization

Consensus-based optimization (CBO) is an agent-based, derivative-free approach for non-smooth global optimization that was introduced in 2017. It utilizes a remarkable interplay between stochastic exploration and Laplace’s principle. CBO has shown its efficacy in various applications and is also well-suited for theoretical convergence analysis. In this talk, I will begin by reviewing the CBO method, followed by a presentation of results regarding the well-posedness of the associated degenerate non-linear Fokker-Planck equation. Finally, I will introduce a variant of CBO designed to address non-convex optimization problems with multiple global minima, demonstrating the existence of solutions for the corresponding nonlinear Fokker-Planck equation, as well as exponential concentration over time towards a set of minimizers consisting of multiple smooth, convex, and compact components.