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

Jérôme Bolte, Laurent Miclo () and Stéphane Villeneuve
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Laurent Miclo: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Stéphane Villeneuve: TSM - Toulouse School of Management Research - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - CNRS - Centre National de la Recherche Scientifique - TSM - Toulouse School of Management - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse

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Abstract: 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.

Date: 2024
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Published in Mathematical Programming, 2024, 205, pp.661-701. ⟨10.1007/s10107-023-01988-8⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-04552722

DOI: 10.1007/s10107-023-01988-8

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