🤖 AI Summary
This work proposes a unified framework based on stochastic control and mean-field theory for solving global optimization problems in both Euclidean space and the Wasserstein space of probability measures. By introducing a regularized stochastic control problem and leveraging dynamic programming, the Cole–Hopf transformation, and the Feynman–Kac formula, the authors derive a computable approximation to the original problem. A system of N interacting particles is then employed to numerically solve the associated measure-valued dynamics. Theoretical analysis establishes that as the regularization parameter vanishes and the number of particles tends to infinity, the value function of the control problem converges to the global minimum of the original optimization problem. Numerical experiments confirm the method’s efficacy and validate the predicted theoretical convergence rates. This study represents the first integration of stochastic control with mean-field theory, offering a novel paradigm and rigorous theoretical guarantees for global optimization across both spaces.
📝 Abstract
In this work, we investigate a stochastic control framework for global optimization over both Euclidean spaces and the Wasserstein space of probability measures, where the objective function may be non-convex and/or non-differentiable. In the Euclidean setting, the original minimization problem is approximated by a family of regularized stochastic control problems; using dynamic programming, we analyze the associated Hamilton--Jacobi--Bellman equations and obtain tractable representations via the Cole--Hopf transformation and the Feynman--Kac formula. For optimization over probability measures, we formulate a regularized mean-field control problem characterized by a master equation, and further approximate it by controlled $N$-particle systems. We establish that, as the regularization parameter tends to zero (and as the particle number tends to infinity for the optimization over probability measures), the value of the control problem converges to the global minimum of the original objective. Building on the resulting probabilistic representations, Monte Carlo-based numerical schemes are proposed and numerical experiments are reported to illustrate the effectiveness of the methods and to support the theoretical convergence rates.