Convergence rates in forward-backward splitting

Forward-backward splitting methods provide a range of approaches to solving large-scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized. Apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out, efforts at evaluating the convergence potential of such methods have so far relied on Lipschitz properties and strong monotonicity, or inverse strong monotonicity, of the mapping involved in the forward step, the perspective mainly being that of projection algorithms. Here, convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well. For the first time in such a general setting, global and local contraction rates are derived; moreover, they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem. Insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected.