Swarm gradient dynamics for global optimization: the density case

TL;DR

This paper introduces swarm gradient dynamics, a novel global optimization framework combining geometric, stochastic, and gradient-based methods, with theoretical guarantees in one-dimensional cases and potential for broader applications.

Contribution

It extends simulated annealing to a new class of swarm gradient methods and proves a key functional inequality for one-dimensional manifolds, suggesting wider applicability.

Findings

Proved a functional inequality for one-dimensional compact manifolds.

Established convergence conditions for swarm gradient dynamics.

Highlighted the importance of functional inequalities like Łojasiewicz in optimization.

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 {\`a} la {\L}ojasiewicz.