  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.
