Constraint Coupled Distributed Optimization: Relaxation and Duality Approach

In this paper we consider a distributed optimization scenario in which agents of a network want to minimize the sum of local convex cost functions, each one depending on a local variable, subject to convex local and coupling constraints, with the latter involving all the decision variables. We propose a novel 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 linearly updates other dual variables. We prove that agents’ estimates asymptotically converge to an optimal solution of the given problem, namely to a point satisfying both local and coupling constraints and having optimal cost. This primal recovery property is obtained without any averaging mechanism typically used in dual 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.