  This work develops a proximal primal-dual decentralized strategy for
multi-agent optimization problems that involve multiple coupled affine
constraints, where each constraint may involve only a subset of the agents. The
constraints are generally sparse, meaning that only a small subset of the
agents are involved in them. This scenario arises in many applications
including decentralized control formulations, resource allocation problems, and
smart grids. Traditional decentralized solutions tend to ignore the structure
of the constraints and lead to degraded performance. We instead develop a
decentralized solution that exploits the sparsity structure. Under constant
step-size learning, the asymptotic convergence of the proposed algorithm is
established in the presence of non-smooth terms, and it occurs at a linear rate
in the smooth case. We also examine how the performance of the algorithm is
influenced by the sparsity of the constraints. Simulations illustrate the
superior performance of the proposed strategy.
