Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators

Abstract : We investigate the asymptotic behavior of a stochastic version of the forward-backward splitting algorithm for finding a zero of the sum of a maximally monotone set-valued operator and a cocoercive operator in Hilbert spaces. Our general setting features stochastic approximations of the cocoercive operator and stochastic perturbations in the evaluation of the resolvents of the set-valued operator. In addition, relaxations and not necessarily vanishing proximal parameters are allowed. Weak and strong almost sure convergence properties of the iterates is established under mild conditions on the underlying stochastic processes. Leveraging these results, we also establish the almost sure convergence of the iterates of a stochastic variant of a primal-dual proximal splitting method for composite minimization problems.
Type de document :
Article dans une revue
Pure and Applied Functional Analysis, 2016, 1 (1), pp.13-37. <http://www.ybook.co.jp/online2/oppafa/vol1/p13.html>
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https://hal.archives-ouvertes.fr/hal-01380000
Contributeur : Emilie Chouzenoux <>
Soumis le : mercredi 12 octobre 2016 - 12:27:29
Dernière modification le : mercredi 15 février 2017 - 13:52:12

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  • HAL Id : hal-01380000, version 1

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Patrick Louis Combettes, Jean-Christophe Pesquet. Stochastic Approximations and Perturbations in Forward-Backward Splitting for Monotone Operators. Pure and Applied Functional Analysis, 2016, 1 (1), pp.13-37. <http://www.ybook.co.jp/online2/oppafa/vol1/p13.html>. <hal-01380000>

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