  We consider the problem of minimizing a convex, separable, nonsmooth function
subject to linear constraints. The numerical method we propose is a
block-coordinate extension of the Chambolle-Pock primal-dual algorithm. We
prove convergence of the method without resorting to assumptions like
smoothness or strong convexity of the objective, full-rank condition on the
matrix, strong duality or even consistency of the linear system. Freedom from
imposing the latter assumption permits convergence guarantees for misspecified
or noisy systems.
