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.
In the same vein as our paper, @cite_28 considered a collection @math of @math maximal monotone operators, and studied the iterations [ y_ n+1 A( n+1 (1), x_n) , , x_ n+1 = i=2 ^N (I + n+1 A( n+1 (i), ))^ -1 (x_n - n+1 y_ n+1 ) , , ] where @math , and where @math is a sequence of permutations of the set @math . The convergence of @math to a zero of @math is established in @cite_28 . In the recent paper @cite_12 , a relaxed version of Algorithm is considered, where @math is cocoercive and where its output, as well as the output of the resolvent of @math , are subjected to random errors. The convergence of the iterates to a zero of @math is established under summability assumptions on these errors.
@cite_28 @cite_12
1998576973 2963692017
In a method of treating a textile material in a shrinking bath containing a swelling agent comprising the step of introducing into said bath a driving-off agent which substantially reduces the solubility of the swelling agent in the shrinking bath. 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.