  In this work, we show that for linearly constrained optimization problems the
primal-dual hybrid gradient algorithm, analyzed by Chambolle and Pock [3], can
be written as an entirely primal algorithm. This allows us to prove convergence
of the iterates even in the degenerate cases when the linear system is
inconsistent or when the strong duality does not hold. We also obtain new
convergence rates which seem to improve existing ones in the literature. For a
decentralized distributed optimization we show that the new scheme is much more
efficient than the original one.
