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

In this paper we study the relaxed primal-dual algorithm for solving composite monotone inclusions in real Hilbert spaces with critical preconditioners. Our approach is based in new results on the asymptotic behaviour of Krasnosel'ski\u{\i}-Mann (KM) iterations defined in the range of monotone self-adjoint linear operators. These results generalize the convergence of classical KM iterations aiming at approximating fixed points. We prove that the relaxed primal-dual algorithm with critical preconditioners define KM iterations in the range of a particular monotone self-adjoint linear operator with non-trivial kernel. We then deduce from our fixed point approach that the shadows of primal-dual iterates on the range of the linear operator converges weakly to some point in this vector subspace from which we obtain a solution. This generalizes (Condat 2013 Theorem 3.3) to infinite dimensional relaxed primal-dual monotone inclusions involving critical preconditioners. The Douglas-Rachford splitting (DRS) is interpreted as a particular instance of the primal-dual algorithm when the step-sizes are critical and we recover classical results from this new perspective. We implement the relaxed primal-dual algorithm with critical preconditioners in total variation reconstruction and we illustrate its flexibility and efficiency.