  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.
