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

The primal-dual hybrid gradient method reduces to a primal method for linearly constrained optimization problems