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

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

“…be the set of distributions π such that π = πP γ for at least one P γ with γ ∈ (0, γ 0 ]. The following proposition, which is valid for Feller Markov kernels, has been proven in [13] in the more general context of set-valued differential inclusions.…”

“…3.3 are satisfied. Namely, we need to establish (13) and to show that the cluster points of I(P) as γ → 0 are elements of I(Φ).…”

“…This case is considered in [19] (see also [18]), which relies on a Robbins-Siegmund like approach requiring summability assumptions on the random errors. The constant step case is also dealt with in [39] and in [13] for generic differential inclusions. In the present work, we follow the line of reasoning of our paper [13], noting that the case where the DI is defined by a maximal monotone operator has many specificities.…”

“…The constant step case is also dealt with in [39] and in [13] for generic differential inclusions. In the present work, we follow the line of reasoning of our paper [13], noting that the case where the DI is defined by a maximal monotone operator has many specificities. For instance, a maximal monotone operator is not upper semi continuous in general, as it was assumed for the differential inclusions studied in [39] and [13].…”

“…In the present work, we follow the line of reasoning of our paper [13], noting that the case where the DI is defined by a maximal monotone operator has many specificities. For instance, a maximal monotone operator is not upper semi continuous in general, as it was assumed for the differential inclusions studied in [39] and [13]. Another difference lies in the fact that we consider here the case where the domains of the operators A(s) can be different.…”