  Consensus-based optimization (CBO) is an agent-based derivative-free method
for non-smooth global optimization that has been introduced in 2017, leveraging
a surprising interplay between stochastic exploration and Laplace principle. In
addition to its versatility and effectiveness in handling high-dimensional,
non-convex, and nonsmooth optimization problems, this approach lends itself
well to theoretical analysis. Indeed, its dynamics is governed by a degenerate
nonlinear Fokker-Planck equation, whose large-time behavior explains the
convergence of the method. Recent results provide guarantees of convergence
under the restrictive assumption of a unique global minimizer for the objective
function. In this work, we proposed a novel and simple variation of CBO to
tackle nonconvex optimizations with multiple minimizers. Despite the simplicity
of this new model, its analysis is particularly challenging because of its
nonlinearity and nonlocal nature. We prove the existence of solutions of the
corresponding nonlinear Fokker-Planck equation and we show exponential
concentration in time to the set of minimizers made of multiple smooth, convex,
and compact components. Our proofs require combining several ingredients, such
as delicate geometrical arguments, new variants of a quantitative Laplace
principle, ad hoc regularizations and approximations, and regularity theory for
parabolic equations. Ultimately, this result suggests that the corresponding
CBO algorithm, formulated as an Euler-Maruyama discretization of the underlying
empirical stochastic process, tends to converge to global minimizers.
