Primal-dual splittings as fixed point iterations in the range of linear operators

In this paper we study the convergence of the relaxed primal-dual algorithm with critical preconditioners for solving composite monotone inclusions in real Hilbert spaces. We prove that this algorithm define Krasnosel'skii-Mann (KM) iterations in the range of a particular monotone self-adjoint linear operator with non-trivial kernel. Our convergence result generalizes (Condat in J Optim Theory Appl 158: 460-479, 2013, Theorem 3.3) and follows from that of KM iterations defined in the range of linear operators, which is a real Hilbert subspace under suitable conditions. The Douglas-Rachford splitting (DRS) with a non-standard metric is written as a particular instance of the primal-dual algorithm with critical preconditioners and we recover classical results from this new perspective. We implement the algorithm in total variation reconstruction, verifying the advantages of using critical preconditioners and relaxation steps.