optimal cost $f^\star := \sum_{p=1}^P f_p(x^\star)$ is finite (and that $\cap_{p=1}^P X_p \neq \emptyset$). In Assumption \[Ass:Network\], a network is connected if there is a path between every pair of nodes.

Under the previous assumptions, we solve the following problem: *given a network, design a distributed algorithm that solves *. By distributed algorithm, we mean that there is no notion of a central or special node; also, no node in the network, except node $p$, has access to $f_p$ or $X_p$ at any time before or during the algorithm.

[[**Related work.**]{}]{} The first method addressing the generic problem  was based on an average consensus algorithm coupled with a subgradient method [@Lobel10-DistributedMultiAgentOptimization; @CXVOptimInSPandComm]. Such method has been extensively studied and is known to be robust to noise and link failures; however, it generally requires too many iterations to converge.

Another approach for solving  in a distributed way is via augmented Lagrangian methods, which consist of two loops: an outer loop updating the dual variables, and an inner loop updating the primal variables. There are some options for the algorithms in each loop. For the outer loop, the most common is the gradient method [@Ruszczynski; @Spars09; @Jakovetic11-CooperativeConvexOptimization], which degenerates in the so-called 