Constant Step Stochastic Approximations Involving Differential Inclusions: Stability, Long-Run Convergence and Applications

We consider a Markov chain $(x_n)$ whose kernel is indexed by a scaling parameter $\gamma>0$, refered to as the step size. The aim is to analyze the behavior of the Markov chain in the doubly asymptotic regime where $n\to\infty$ then $\gamma\to 0$. First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putting the above results together, we establish the long run convergence of the iterates as $\gamma\to 0$, to the Birkhoff center of the DI. The ergodic behavior of the iterates is also provided. Application examples are investigated. We apply our findings to 1) the problem of nonconvex proximal stochastic optimization and 2) a fluid model of parallel queues.