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

Jérôme Bolte (1) , Laurent Miclo (1) , Stéphane Villeneuve (2)

1 TSE-R - Toulouse School of Economics
2 TSM - Toulouse School of Management Research

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.

Domaines: Economies et finances; Optimisation et contrôle [math.OC]

Jérôme Bolte, Laurent Miclo, Stéphane Villeneuve. Swarm gradient dynamics for global optimization: the mean-field limit case. Mathematical Programming, 2024, 205, pp.661-701. ⟨10.1007/s10107-023-01988-8⟩. ⟨hal-04552722⟩