Constraint Coupled Distributed Optimization: a Relaxation and Duality Approach

TL;DR

This paper introduces a novel distributed optimization algorithm for network control problems with local and coupling constraints, leveraging relaxation and duality theory to ensure convergence to optimal solutions without averaging.

Contribution

It presents a fully distributed primal-dual algorithm based on problem relaxation and duality, with proven convergence to feasible optimal solutions in network optimization.

Findings

Algorithm converges asymptotically to an optimal solution.

No averaging mechanism needed for primal recovery.

Applicable to distributed control in microgrid scenarios.

Abstract

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.