Well-posedness and approximation of reflected McKean-Vlasov SDEs with applications

TL;DR

This work develops a rigorous framework for reflected McKean–Vlasov SDEs in smooth non-convex domains, proving well-posedness for the mean-field equation and its particle approximation. It then analyzes the convergence of the interacting-particle system to the mean-field limit at optimal rates and studies long-time behavior using reflection coupling, including additive-noise scenarios. The authors instantiate the theory with reflected mean-field Langevin dynamics and two constrained consensus-based optimization models—one augmented with attracting/repelling forces—establishing convergence to global minima under appropriate conditions. Numerical experiments on constrained optimization problems and an inverse problem demonstrate the practical viability of the reflected SDE framework and its discretizations. Overall, the paper provides a comprehensive theory plus practical algorithms for sampling and optimization under domain constraints using mean-field and particle methods.

Abstract

In this paper, we establish well-posedness of reflected McKean-Vlasov SDEs and their particle approximations in smooth non-convex domains. We prove convergence of the interacting particle system to the corresponding mean-field limit with the optimal rate of convergence. We motivate this study with applications to sampling and optimization in constrained domains by considering reflected mean-field Langevin SDEs and two reflected consensus-based optimization (CBO) models, respectively. We utilize reflection coupling to study long-time behaviour of reflected mean-field SDEs and also investigate convergence of the reflected CBO models to the global minimum of a constrained optimization problem. We numerically test reflected CBO models on benchmark constrained optimization problems and an inverse problem.

Figures (5)

• Figure 5.1: The example of a non-convex domain, where the Euclidean path of a particle to the consensus does not lie within the domain. In the convex case, this behavior cannot occur.
• Figure 6.1: Translated Ackley function with minimum at $(2, 2)$ constrained to a closed ball of radius 3.
• Figure 6.2: Non-convex function with heart-shaped constraint.
• Figure 6.3: Rosenbrock function constrained to a closed ball of radius $\sqrt{2}$.
• Figure 6.4: The inverse problem. Histograms of recovered parameters across the 1000 experiments.

Theorems & Definitions (21)

• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:wp_mf']}
• Lemma 3.1
• proof : Proof of Lemma \ref{['lemma:contraction']}
• Theorem 3.2
• proof
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 4.1