Random iterations of paracontraction maps and applications to feasibility problems

TL;DR

This paper proves that random iterations of paracontraction maps converge almost surely to a common fixed point, providing new stochastic algorithms for feasibility and monotone operator problems.

Contribution

It introduces a convergence result for stochastic iterations of paracontraction maps and applies it to develop randomized algorithms for feasibility and monotone operator problems.

Findings

Random iteration of paracontraction maps converges almost surely.

Applications to stochastic convex feasibility problems.

Development of non-white noise randomized algorithms.

Abstract

In this paper, we consider the problem of finding an almost surely common fixed point of a family of paracontraction maps indexed on a probability space, which we refer to as the stochastic feasibility problem. We show that a random iteration of paracontraction maps driven by an ergodic stationary sequence converges, with probability one, to a solution of the stochastic feasibility problem, provided a solution exists. As applications, we obtain non-white noise randomized algorithms to solve the stochastic convex feasibility problem and the problem of finding an almost surely common zero of a collection of maximal monotone operators.