  We study distributed optimization in a cooperative multi-agent setting, where
agents have to agree on the usage of shared resources and can communicate via a
time-varying network to this purpose. Each agent has its own decision variables
that should be set so as to minimize its individual objective function subject
to local constraints. Resource sharing is modeled via coupling constraints that
involve the non-positivity of the sum of agents' individual functions, each one
depending on the decision variables of one single agent. We propose a novel
distributed algorithm to minimize the sum of the agents' objective functions
subject to both local and coupling constraints, where dual decomposition and
proximal minimization are combined in an iterative scheme. Notably, privacy of
information is guaranteed since only the dual optimization variables associated
with the coupling constraints are exchanged by the agents. Under convexity
assumptions, jointly with suitable connectivity properties of the communication
network, we are able to prove that agents reach consensus to some optimal
solution of the centralized dual problem counterpart, while primal variables
converge to the set of optimizers of the centralized primal problem. The
efficacy of the proposed approach is demonstrated on a plug-in electric
vehicles charging problem.
