2022/04/04 by Jérôme Bolte, Laurent Miclo, Bolte, Jérôme +3
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Optimization and Variational Analysis #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2204.01306
openalex publication_date 2022/04/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.