This work studies a class of non-smooth decentralized multi-agent optimization problems where the agents aim at minimizing a sum of local strongly-convex smooth components plus a common non-smooth term. We propose a general primal-dual algorithmic framework that unifies many existing state-of-the-art algorithms. We establish linear convergence of the proposed method to the exact solution in the presence of the non-smooth term. Moreover, for the more general class of problems with agent specific non-smooth terms, we show that linear convergence cannot be achieved (in the worst case) for the class of algorithms that uses the gradients and the proximal mappings of the smooth and non-smooth parts, respectively. We further provide a numerical counterexample that shows how some state-of-the-art algorithms fail to converge linearly for strongly-convex objectives and different local non-smooth terms.
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