  In this paper we consider a general, challenging distributed optimization
set-up arising in several important network control applications. Agents of a
network want to minimize the sum of local cost functions, each one depending on
a local variable, subject to local and coupling constraints, with the latter
involving all the decision variables. We propose a novel fully distributed
algorithm based on a relaxation of the primal problem and an elegant
exploration of duality theory. Despite its complex derivation, based on several
duality steps, the distributed algorithm has a very simple and intuitive
structure. That is, each node finds a primal-dual optimal solution pair of a
local, relaxed version of the original problem, and then updates suitable
auxiliary local variables. We prove that agents asymptotically compute their
portion of an optimal (feasible) solution of the original problem. This primal
recovery property is obtained without any averaging mechanism typically used in
dual decomposition methods. To corroborate the theoretical results, we show how
the methodology applies to an instance of a Distributed Model Predictive
Control scheme in a microgrid control scenario.
