  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.
