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

Author

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  • Jérôme Bolte

    (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)

  • 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)

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.

Suggested Citation

  • Jérôme Bolte & Laurent Miclo & Stéphane Villeneuve, 2024. "Swarm gradient dynamics for global optimization: the mean-field limit case," Post-Print hal-04552722, HAL.
    • Handle: RePEc:hal:journl:hal-04552722
    DOI: 10.1007/s10107-023-01988-8
    Note: View the original document on HAL open archive server: https://hal.science/hal-04552722v1
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    References listed on IDEAS

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    1. Arnak S. Dalalyan, 2017. "Theoretical guarantees for approximate sampling from smooth and log-concave densities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 651-676, June.
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