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

Pascal Bianchi () and Walid Hachem ()
Additional contact information
Pascal Bianchi: Université Paris-Saclay
Walid Hachem: Université Paris-Saclay

Journal of Optimization Theory and Applications, 2016, vol. 171, issue 1, No 5, 90-120

Abstract: 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.

Keywords: Dynamical systems; Random maximal monotone operators; Stochastic Forward–Backward algorithm; Stochastic proximal point algorithm; 47H05; 47N10; 62L20; 34A60 (search for similar items in EconPapers)
Date: 2016
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://link.springer.com/10.1007/s10957-016-0978-y Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:171:y:2016:i:1:d:10.1007_s10957-016-0978-y

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-016-0978-y

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().