Convergence rates of Forward–Douglas–Rachford splitting method

Over the past decades, operator splitting methods have become ubiquitous for nonsmooth optimization owing to their simplicity and efficiency. In this paper, we consider the Forward–Douglas–Rachford splitting method and study both global and local convergence rates of this method. For the global rate, we establish a sublinear convergence rate in terms of a Bregman divergence suitably designed for the objective function. Moreover, when specializing to the Forward–Backward splitting, we prove a stronger convergence rate result for the objective function value. Then locally, based on the assumption that the non-smooth part of the optimization problem is partly smooth, we establish local linear convergence of the method. More precisely, we show that the sequence generated by Forward–Douglas–Rachford first (i) identifies a smooth manifold in a finite number of iteration and then (ii) enters a local linear convergence regime, which is for instance characterized in terms of the structure of the underlying active smooth manifold. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from applicative fields including, for instance, signal/image processing, inverse problems and machine learning