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 pseudo trajectory in the sense of Bena " m and Hirsch of the differential inclusion involving the sum of the mean operators. The convergence of the empirical means of the iterates towards 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 or variational inequality problems in random environments.
The purpose of this paper is to establish the almost sure weak ergodic convergence of a sequence of iterates @math given by @math where @math is a collection of maximal monotone operators on a separable Hilbert space, @math is an independent identically distributed sequence of random variables on @math and @math is a positive sequence in @math . The weighted averaged sequence of iterates is shown to converge weakly to a zero (assumed to exist) of the Aumann expectation @math under the assumption that the latter is maximal. We consider applications to stochastic optimization problems of the form @math , where @math is a normal convex integrand and @math is a collection of closed convex sets. In this case, the iterations are closely related to a stochastic proximal algorithm recently proposed by Wang and Bertsekas [Increment...