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

• 成果类型: Article
• 署名作者: Bolte, Jerome; Miclo, Laurent; Villeneuve, Stephane
• 署名单位: Universite de Toulouse; Universite Toulouse 1 Capitole; Toulouse School of Economics
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01988-8
• 发表日期: 2024
• 页码: 661-701
• 关键词: entropy dissipation; EQUATIONS; inequalities; EVOLUTION

摘要

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 Lojasiewicz.