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

Abstract

Using jointly geometric and stochastic reformulations of nonconvex problems and exploiting a Monge-Kantorovich gradient system formulation with vanishing forces, we formally extend the simulated annealing method to a wide class of global optimization methods. Due to an inbuilt combination of a gradient-like strategy and particles interactions, we call them swarm gradient dynamics. As in the original paper of Holley-Kusuoka- Stroock, the key to the existence of a schedule ensuring convergence to a global minimizer is a functional inequality. One of our central theoretical contributions is the proof of such an inequality for one-dimensional compact manifolds. We conjecture the inequality to be true in a much wider setting. We also describe a general method allowing for global optimization and evidencing the crucial role of functional inequalities à la Łojasiewicz.

Replaces

Jérôme Bolte, Laurent Miclo, and Stéphane Villeneuve, “Swarm gradient dynamics for global optimization: the mean-field limit case”, TSE Working Paper, n. 22-1302, March 2022.

Reference

Jérôme Bolte, Laurent Miclo, and Stéphane Villeneuve, “Swarm gradient dynamics for global optimization: the mean-field limit case”, Mathematical Programming, 2024, forthcoming.