Degenerate Preconditioned Proximal Point algorithms

In this paper we describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak convergence results under mild assumptions that can be easily employed in the context of splitting methods for monotone inclusion and convex minimization problems. Moreover, we show that the degeneracy of the preconditioner allows for a reduction of the variables involved in the iteration updates. We show the strength of the proposed framework in the context of splitting algorithms, providing new simplified proofs of convergence and highlighting the link between existing schemes, such as Chambolle-Pock, Forward Douglas-Rachford and Peaceman-Rachford, that we study from a preconditioned proximal point perspective. The proposed framework allows to devise new flexible schemes and provides new ways to generalize existing splitting schemes to the case of the sum of many terms. As an example, we present a new sequential generalization of Forward Douglas-Rachford along with numerical experiments that demonstrate its interest in the context of nonsmooth convex optimization.