An alternative approach to well-posedness of McKean-Vlasov equations arising in Consensus-Based Optimization

TL;DR

This paper introduces a new method to prove the well-posedness of McKean-Vlasov equations in Consensus-Based Optimization by using a truncation approach with a cut-off function, extending existing results.

Contribution

It presents a novel truncation-based approach to establish well-posedness of non-Lipschitz McKean-Vlasov equations in CBO, broadening the class of solutions.

Findings

Established existence of strong solutions.

Extended pathwise uniqueness to a larger class of solutions.

Provided a new framework for analyzing non-Lipschitz McKean-Vlasov equations.

Abstract

In this work we study the mean-field description of Consensus-Based Optimization (CBO), a derivative-free particle optimization method. Such a description is provided by a non-local SDE of McKean-Vlasov type, whose fields lack of global Lipschitz continuity. We propose a novel approach to prove the well-posedness of the mean-field CBO equation based on a truncation argument. The latter is performed through the introduction of a cut-off function, defined on the space of probability measures, acting on the fields. This procedure allows us to study the well-posedness problem in the classical framework of Sznitman. Through this argument, we recover the established result on the existence of strong solutions, and we extend the class of solutions for which pathwise uniqueness holds.