Swarm gradient dynamics for global optimization: the mean-field limit case

Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge–Kantorovich (or Wasserstein) gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide range of global optimization methods. Due to the built-in combination of a gradient-like strategy and particle interactions, we call them swarm gradient dynamics. As in the original paper by Holley–Kusuoka–Stroock, a functional inequality is the key to the existence of a schedule that ensures convergence to a global minimizer. One of our central theoretical contributions is proving such an inequality for one-dimensional compact manifolds. We conjecture that the inequality holds true in a much broader setting. Additionally, we describe a general method for global optimization that highlights the essential role of functional inequalities la Łojasiewicz.