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Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators

Author

Listed:
  • Pascal Bianchi

    (Université Paris-Saclay)

  • Walid Hachem

    (Université Paris-Saclay)

Abstract

The purpose of this paper is to study the dynamical behavior of the sequence produced by a Forward–Backward algorithm, involving two random maximal monotone operators and a sequence of decreasing step sizes. Defining a mean monotone operator as an Aumann integral and assuming that the sum of the two mean operators is maximal (sufficient maximality conditions are provided), it is shown that with probability one, the interpolated process obtained from the iterates is an asymptotic pseudotrajectory in the sense of Benaïm and Hirsch of the differential inclusion involving the sum of the mean operators. The convergence of the empirical means of the iterates toward a zero of the sum of the mean operators is shown, as well as the convergence of the sequence itself to such a zero under a demipositivity assumption. These results find applications in a wide range of optimization problems or variational inequalities in random environments.

Suggested Citation

  • Pascal Bianchi & Walid Hachem, 2016. "Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 90-120, October.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0978-y
    DOI: 10.1007/s10957-016-0978-y
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    References listed on IDEAS

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    1. Michel Benaim & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions II: Applications," Levine's Bibliography 784828000000000098, UCLA Department of Economics.
    2. Michel Benaïm & Josef Hofbauer & Sylvain Sorin, 2005. "Stochastic Approximations and Differential Inclusions; Part II: Applications," Working Papers hal-00242974, HAL.
    3. Hiai, Fumio & Umegaki, Hisaharu, 1977. "Integrals, conditional expectations, and martingales of multivalued functions," Journal of Multivariate Analysis, Elsevier, vol. 7(1), pages 149-182, March.
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    Cited by:

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    2. Panos Toulis & Thibaut Horel & Edoardo M. Airoldi, 2021. "The proximal Robbins–Monro method," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(1), pages 188-212, February.
    3. Andrés Contreras & Juan Peypouquet, 2019. "Asymptotic Equivalence of Evolution Equations Governed by Cocoercive Operators and Their Forward Discretizations," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 30-48, July.

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