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.

Abstract-We provide a unifying framework for distributed convex optimization over time-varying networks, in the presence of constraints and uncertainty, features that are typically treated separately in the literature. We adopt a proximal minimization perspective and show that this set-up allows us to bypass the difficulties of existing algorithms while simplifying the underlying mathematical analysis. We develop an iterative algorithm and show convergence of the resulting scheme to some optimizer of the centralized problem. To deal with the case where the agents' constraint sets are affected by a possibly common uncertainty vector, we follow a scenario-based methodology and offer probabilistic guarantees regarding the feasibility properties of the resulting solution. To this end, we provide a distributed implementation of the scenario approach, allowing agents to use a different set of uncertainty scenarios in their local optimization programs. The efficacy of our algorithm is demonstrated by means of a numerical example related to a regression problem subject to regularization.

The identification of polynomial Nonlinear Autoregressive [Moving Average] models with eXogenous variables (NAR[MA]X) is typically carried out with incremental model building techniques that progressively select the terms to include in the model. The Model Structure Selection (MSS) turns out to be the hardest task of the identification process due to the difficulty of correctly evaluating the importance of a generic term. As a result, classical MSS methods sometimes yield unsatisfactory models, that are unreliable over long-range prediction horizons. The MSS problem is here recast into a probabilistic framework based on which a randomized algorithm for MSS is derived, denoted RaMSS. The method introduces a tentative probability distribution over models and progressively updates it by extracting useful information on the importance of each term from sampled model structures. The proposed method is validated over models with different characteristics by means of Monte Carlo simulations, which show its advantages over classical and competitor probabilistic MSS methods in terms of both reliability and computational efficiency

We consider constraint-coupled optimization problems in which agents of a network aim to cooperatively minimize the sum of local objective functions subject to individual constraints and a common linear coupling constraint. We propose a novel optimization algorithm that embeds a dynamic average consensus protocol in the parallel Alternating Direction Method of Multipliers (ADMM) to design a fully distributed scheme for the considered set-up. The dynamic average mechanism allows agents to track the time-varying coupling constraint violation (at the current solution estimates). The tracked version of the constraint violation is then used to update local dual variables in a consensus-based scheme mimicking a parallel ADMM step. Under convexity, we prove that all limit points of the agents' primal solution estimates form an optimal solution of the constraint-coupled (primal) problem. The result is proved by means of a Lyapunov-based analysis simultaneously showing consensus of the dual estimates to a dual optimal solution, convergence of the tracking scheme and asymptotic optimality of primal iterates. A numerical study on optimal charging schedule of plug-in electric vehicles corroborates the theoretical results.

We address the optimal design of a large scale multi-agent system where each agent has discrete and/or continuous decision variables that need to be set so as to optimize the sum of linear local cost functions, in presence of linear local and global constraints. The problem reduces to a Mixed Integer Linear Program (MILP) that is here addressed according to a decentralized iterative scheme based on dual decomposition, where each agent determines its decision vector by solving a smaller MILP involving its local cost function and constraint given some dual variable, whereas a central unit enforces the global coupling constraint by updating the dual variable based on the tentative primal solutions of all agents. An appropriate tightening of the coupling constraint through iterations allows to obtain a solution that is feasible for the original MILP. The proposed approach is inspired by a recent method to the MILP approximate solution via dual decomposition and constraint tightening, and presents the advantage of guaranteeing feasibility in finite-time and providing better performance guarantees. The two approaches are compared on a numerical example on plug-in electric vehicles optimal charging.