Convergence of a relaxed inertial proximal algorithm for maximally monotone operators

Abstract

In a Hilbert space $${\mathcal {H}}$$ , given $$A{:}\;{\mathcal {H}}\rightarrow 2^{\mathcal {H}}$$ a maximally monotone operator, we study the convergence properties of a general class of relaxed inertial proximal algorithms. This study aims to extend to the case of the general monotone inclusion $$Ax \ni 0$$ the acceleration techniques initially introduced by Nesterov in the case of convex minimization. The relaxed form of the proximal algorithms plays a central role. It comes naturally with the regularization of the operator A by its Yosida approximation with a variable parameter, a technique recently introduced by Attouch–Peypouquet (Math Program Ser B, 2018. https://doi.org/10.1007/s10107-018-1252-x ) for a particular class of inertial proximal algorithms. Our study provides an algorithmic version of the convergence results obtained by Attouch–Cabot (J Differ Equ 264:7138–7182, 2018) in the case of continuous dynamical systems.