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In this paper we consider a network scenario in which agents can evaluate each other according to a score graph that models some physical or social interaction. The goal is to design a distributed protocol, run by the agents, allowing them to learn their unknown state among a finite set of possible values. We propose a Bayesian framework in which scores and states are associated to probabilistic events with unknown parameters and hyperparameters respectively. We prove that each agent can learn its state by means of a local Bayesian classifier and a (centralized) Maximum-Likelihood (ML) estimator of the parameter-hyperparameter that combines plain ML and Empirical Bayes approaches. By using tools from graphical models, which allow us to gain insight on conditional dependences of scores and states, we provide two relaxed probabilistic models that ultimately lead to ML parameter-hyperparameter estimators amenable to distributed computation. In order to highlight the appropriateness of the proposed relaxations, we demonstrate the distributed estimators on a machine-to-machine testing set-up for anomaly detection and on a social interaction set-up for user profiling.

In this paper, we consider a network of processors aiming at cooperatively solving mixed-integer convex programs subject to uncertainty. Each node only knows a common cost function and its local uncertain constraint set. We propose a randomized, distributed algorithm working under asynchronous, unreliable and directed communication. The algorithm is based on a local computation and communication paradigm. At each communication round, nodes perform two updates: (i) a verification in which they check---in a randomized fashion---the robust feasibility of a candidate optimal point, and (ii) an optimization step in which they exchange their candidate basis (the minimal set of constraints defining a solution) with neighbors and locally solve an optimization problem. As main result, we show that processors can stop the algorithm after a finite number of communication rounds (either because verification has been successful for a sufficient number of rounds or because a given threshold has been reached), so that candidate optimal solutions are consensual. The common solution is proven to be---with high confidence---feasible and hence optimal for the entire set of uncertainty except a subset having an arbitrary small probability measure. We show the effectiveness of the proposed distributed algorithm using two examples: a random, uncertain mixed-integer linear program and a distributed localization in wireless sensor networks. The distributed algorithm is implemented on a multi-core platform in which the nodes communicate asynchronously., Comment: Submitted for publication. arXiv admin note: text overlap with arXiv:1706.00488

In this paper, we consider a general challenging distributed optimization setup 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.

This article deals with distributed control of microgrids composed of storages, generators, renewable energy sources, and critical and controllable loads. We consider a stochastic formulation of the optimal control problem associated with the microgrid that appropriately takes into account the unpredictable nature of the power generated by renewables. The resulting problem is a mixed-integer linear program and is NP-hard and nonconvex. Moreover, the peculiarity of the considered framework is that no central unit can be used to perform the optimization, but rather the units must cooperate with each other by means of neighboring communication. To solve the problem, we resort to a distributed methodology based on a primal decomposition approach. The resulting algorithm is able to compute high-quality feasible solutions to a two-stage stochastic optimization problem, for which we also provide a theoretical upper bound on the constraint violation. Finally, a Monte Carlo numerical computation on a scenario with a large number of devices shows the efficacy of the proposed distributed control approach. The numerical experiments are performed on realistic scenarios obtained from Generative Adversarial Networks (GANs) trained an open-source historical dataset of the EU.

We consider a multi-agent setting with agents exchanging information over a possibly time-varying network, aiming at minimising a separable objective function subject to constraints. To achieve this objective we propose a novel subgradient averaging algorithm that allows for non-differentiable objective functions and different constraint sets per agent. Allowing different constraints per agent simultaneously with a time-varying communication network constitutes a distinctive feature of our approach, extending existing results on distributed subgradient methods. To highlight the necessity of dealing with a different constraint set within a distributed optimisation context, we analyse a problem instance where an existing algorithm does not exhibit a convergent behaviour if adapted to account for different constraint sets. For our proposed iterative scheme we show asymptotic convergence of the iterates to a minimum of the underlying optimisation problem for step sizes of the form η k + 1 , η > 0 . We also analyse this scheme under a step size choice of η k + 1 , η > 0 , and establish a convergence rate of O ( ln k k ) in objective value. To demonstrate the efficacy of the proposed method, we investigate a robust regression problem and an l 2 regression problem with regularisation .

In this paper we deal with a network of computing agents with local processing and neighboring communication capabilities that aim at solving (without any central unit) a submodular optimization problem. The cost function is the sum of many local submodular functions and each agent in the network has access to one function in the sum only. In this \emph{distributed} set-up, in order to preserve their own privacy, agents communicate with neighbors but do not share their local cost functions. We propose a distributed algorithm in which agents resort to the Lov\`{a}sz extension of their local submodular functions and perform local updates and communications in terms of single blocks of the entire optimization variable. Updates are performed by means of a greedy algorithm which is run only until the selected block is computed, thus resulting in a reduced computational burden. The proposed algorithm is shown to converge in expected value to the optimal cost of the problem, and an approximate solution to the submodular problem is retrieved by a thresholding operation. As an application, we consider a distributed image segmentation problem in which each agent has access only to a portion of the entire image. While agents cannot segment the entire image on their own, they correctly complete the task by cooperating through the proposed distributed algorithm.

In this paper we introduce disropt, a Python package for distributed optimization over networks. We focus on cooperative set-ups in which an optimization problem must be solved by peer-to-peer processors (without central coordinators) that have access only to partial knowledge of the entire problem. To reflect this, agents in disropt are modeled as entities that are initialized with their local knowledge of the problem. Agents then run local routines and communicate with each other to solve the global optimization problem. A simple syntax has been designed to allow for an easy modeling of the problems. The package comes with many distributed optimization algorithms that are already embedded. Moreover, the package provides full-fledged functionalities for communication and local computation, which can be used to design and implement new algorithms. disropt is available at github.com/disropt/disropt under the GPL license, with a complete documentation and many examples.

We present a distributed optimization algorithm for solving online personalized optimization problems over a network of computing and communicating nodes, each of which linked to a specific user. The local objective functions are assumed to have a composite structure and to consist of a known time-varying (engineering) part and an unknown (user-specific) part. Regarding the unknown part, it is assumed to have a known parametric (e.g., quadratic) structure a priori, whose parameters are to be learned along with the evolution of the algorithm. The algorithm is composed of two intertwined components: (i) a dynamic gradient tracking scheme for finding local solution estimates and (ii) a recursive least squares scheme for estimating the unknown parameters via user's noisy feedback on the local solution estimates. The algorithm is shown to exhibit a bounded regret under suitable assumptions. Finally, a numerical example corroborates the theoretical analysis.

In this paper, we consider a multi-objective optimization problem over networks in which agents aim to maximize their own objective function, while satisfying both local and coupling constraints. This set up includes, e.g., the computation of optimal steady states in multi-agent control systems. Since fairness is a key feature required for the solution, we resort to Cooperative Game Theory and search for the Nash bargaining solution among all the efficient (or Pareto optimal) points of a bargaining game. We propose a negotiation mechanism among the agents to compute such a solution in a distributed way. The problem is reformulated as the maximization of a properly weighted sum of the objective functions. The proposed algorithm is then a two step procedure in which local estimates of the Nash bargaining weights are updated online and existing distributed optimization algorithms are applied. The proposed method is formally analyzed for a particular case, while numerical simulations are provided to corroborate the theoretical findings and to demonstrate its efficacy

Editors: Karina von Schuckmann; Pierre-Yves Le Traon.-- Review Editors: Neville Smith (Chair); Ananda Pascual; Samuel Djavidnia; Jean-Pierre Gattuso; Marilaure Grégoire; Glenn Nolan., The authors would like to thank the Institut Cartogràfic i Geològic de Catalunya (ICGC) for providing data. Thanks are due to FCT/MCTES for the financial support to CESAM (UID/AMB/50017/2019), through national funds., Chapter 1: Introduction and the European Environment policy framework.-- CMEMS OSR4, Chapter 2: State, variability and change in the ocean.-- CMEMS OSR4, Chapter 3: Case studies.-- CMEMS OSR4, Chapter 4: Specific events 2018.

We consider a multi-agent setting with agents exchanging information over a network to solve a convex constrained optimisation problem in a distributed manner. We analyse a new algorithm based on local subgradient exchange under undirected time-varying communication. First, we prove asymptotic convergence of the iterates to a minimum of the given optimisation problem for time-varying step-sizes of the form $c(k) = \frac{\eta }{{k + 1}}$, for some η > 0. We then restrict attention to step-size choices $c(k) = \frac{\eta }{{\sqrt {k + 1} }},\eta > 0$, and establish a convergence of $\mathcal{O}\left( {\frac{{\ln (k)}}{{\sqrt k }}} \right)$ in objective value. Our algorithm extends currently available distributed subgradient/proximal methods by: (i) accounting for different constraint sets at each node, and (ii) enhancing the convergence speed thanks to a subgradient averaging step performed by the agents. A numerical example demonstrates the efficacy of the proposed algorithm.

In this paper we deal with distributed optimal control for nonlinear dynamical systems over graph, that is large-scale systems in which the dynamics of each subsystem depends on neighboring states only. Starting from a previous work in which we designed a partially distributed solution based on a cloud, here we propose a fully-distributed algorithm. The key novelty of the approach in this paper is the design of a sparse controller to stabilize trajectories of the nonlinear system at each iteration of the distributed algorithm. The proposed controller is based on the design of a stabilizing controller for polytopic Linear Parameter Varying (LPV) systems satisfying nonconvex sparsity constraints. Thanks to a suitable choice of vertex matrices and to an iterative procedure using convex approximations of the nonconvex matrix problem, we are able to design a controller in which each agent can locally compute the feedback gains at each iteration by simply combining coefficients of some vertex matrices that can be pre-computed offline. We show the effectiveness of the strategy on simulations performed on a multi-agent formation control problem.

We consider a network scenario in which agents can evaluate each other according to a score graph that models some interactions. The goal is to design a distributed protocol, run by the agents, that allows them to learn their unknown state among a finite set of possible values. We propose a Bayesian framework in which scores and states are associated to probabilistic events with unknown parameters and hyperparameters, respectively. We show that each agent can learn its state by means of a local Bayesian classifier and a (centralized) Maximum-Likelihood (ML) estimator of parameter-hyperparameter that combines plain ML and Empirical Bayes approaches. By using tools from graphical models, which allow us to gain insight on conditional dependencies of scores and states, we provide a relaxed probabilistic model that ultimately leads to a parameter-hyperparameter estimator amenable to distributed computation. To highlight the appropriateness of the proposed relaxation, we demonstrate the distributed estimators on a social interaction set-up for user profiling., This submission is a shorter work (for conference publication) of a more comprehensive paper, already submitted as arXiv:1706.04081 (under review for journal publication). In this short submission only one social set-up is considered and only one of the relaxed estimators is proposed. Moreover, the exhaustive analysis, carried out in the longer manuscript, is completely missing in this version

Many problems of interest for cyber-physical network systems can be formulated as mixed-integer linear programs in which the constraints are distributed among the agents. In this paper, we propose a distributed algorithmic framework to solve this class of optimization problems in a peer-to-peer network with no coordinator and with limited computation and communication capabilities. At each communication round, agents locally solve a small linear program, generate suitable cutting planes, and communicate a fixed number of active constraints. Within the distributed framework, we first propose an algorithm that, under the assumption of integer-valued optimal cost, guarantees finite-time convergence to an optimal solution. Second, we propose an algorithm for general problems that provides a suboptimal solution up to a given tolerance in a finite number of communication rounds. Both algorithms work under asynchronous, directed, unreliable networks. Finally, through numerical computations, we analyze the algorithm scalability in terms of the network size. Moreover, for a multi-agent multi-task assignment problem, we show, consistently with the theory, its robustness to packet loss.

In this paper we consider repeated coalitional games with transferable utilities (TU) over networks. Namely, we consider a set of n players that have to distribute among themselves a vector of rewards (one for each player). In our network version there is no coordinator allocating the rewards, but the agents have to agree on a common time-averaged vector by updating the local estimates of the reward vector. The common time-averaged reward vector has to approach a suitable constraint set, called core of the game, that guarantees that no agents benefit from quitting the grand coalition. We propose a doubly (over time and space) averaging distributed algorithm. At every iteration, each agent first computes a weighted average of its own time-averaged estimate and those of his neighbors and then generates a new reward vector in order to drive the time-averaged estimate towards a pre-assigned set. The main contribution of the paper is to prove that under certain assumptions, i) all agents' estimates reach consensus on the true time-averaged reward vector, and ii) the estimates (and thus the true time-averaged reward vector) approach the pre-assigned set. Conditions for this to happen are related to the connectivity over time of the communication topology and to the approachability principle.

L'articolo tesse un elogio dell’amore dell’uomo e della donna e della vita di famiglia. S’alimenta dell’intuizione che quest’amore possa diventare generativo in modo eminente: fa amare d’essere, può rivelare che essere è per amare, tal che se ne possa dire come di un trascendentale. Un assunto entusiastico, cui l’autore perviene scegliendo il linguaggio asciutto di un’analisi fenomenologica delle intenzionalità che li costituiscono, ai loro diversi livelli di realtà e di senso. In avvio si prende atto che nelle società della tarda-modernità esistono forme differenziate di coppia, ognuna sembra dar luogo ad una coppia mista; che non c’è la famiglia, ma una vera e propria polinesia di famiglie. Dall’analisi emerge subito però che l’aspetto sintomatico è forse dato dall’affermazione crescente del single come ideale di vita: dominanza di una Weltanschauung estetizzante, di un tipo umano da ultimo autoreferenziale, agognato a tratti anche da quanti scelgono di sposarsi. La prospettiva complessiva della riflessione è l’antropologia pedagogica, scienza di confine tra una pedagogia fondamentale e una filosofia della persona. Intende il piano del poter essere, del dover essere dell’amore e del ‘famigliare’, segnato dal principio-generosità. In ogni grado, la relazione erotica, il legame coniugale e l’alleanza sponsale - grado il più alto, ché consente l’ingresso nell’universo del sacro – si tratta ogni volta d’un modo d’essere che appare possibilità ulteriore offerta alla libertà, permette nuove conquiste nel viaggio alla scoperta di sé e dell’altro. In questo orizzonte si può comprendere come l’amore di coppia possa diventare cifra dell’esistenza personale; e in che senso la comunità familiare possa custodire la genealogia della persona. Dona ad ogni soggetto una mente, un desiderio; ma soprattutto – quando è definita dalla disponibilità ad accogliere «chi viene sempre da altrove» - apre una chiara intuizione della logica del dono, ritmo originario dell’essere e dell’esistenza.

The tourist district is an important tool for territorial governance of regional tourism. The district is a type of Local System which is characterized by its multidimensionality, as well as the spatial contiguity of traders who belong to it. The territory is a space transformed by social interactions and relations of production: it is therefore the indispensable reference for measuring the impact of tourism. A bottom up approach in the process of identification of the districts is therefore vital. A Balance Score Card-approach is desirable to correctly manage this process. This study is a preliminary analysis derived from direct experience of the Sicilian tourist districts, recently formed. La distrito turístico es un importante instrumento para la gobernanza territorial del turismo regional. El distrito es un tipo de sistema local que se caracteriza por su multidimensionalidad, así como la contigüidad espacial de los operadores que pertenecen a ella. El territorio es un espacio transformado por las interacciones sociales y las relaciones de producción: por lo tanto es la referencia indispensable para medir el impacto del turismo.Por tanto, un enfoque ascendente en el proceso de identificación de los distritos es vital. Es deseable administrar correctamente este proceso una puntuación saldo de su tarjeta enfoque. Este estudio es un análisis preliminar derivado de la experiencia directa de los distritos turísti- cos de Sicilia, de reciente formación.

We study a distributed allocation process where, at each time, every player: i) proposes a new bid based on the average utilities produced up to that time, ii) adjusts such allocations based on the inputs received from its neighbors, and iii) generates and allocates new utilities. The average allocations evolve according to a doubly (over time and space) averaging algorithm. We study conditions under which the average allocations reach consensus to any point within a predefined target set even in the presence of adversarial disturbances. Motivations arise in the context of coalitional games with transferable utilities (TU) where the target set is any set of allocations that makes the grand coalition stable.

In this paper we investigate the observability and reachability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a grid or a torus. More in detail, under suitable conditions on the eigenvalue multiplicity, we provide necessary and sufficient conditions, based on simple algebraic rules from number theory, to characterize all and only the nodes from which the network system is observable (reachable). For any set of observation (leader) nodes, we provide a closed form expression for the unobservable (unreachable) eigenvalues and for the eigenvectors of the unobservable (unreachable) subsystem.

In this paper we propose a new control-oriented design technique to enhance the algorithmic performance of the distributed gradient tracking algorithm. We focus on a scenario in which agents in a network aim to cooperatively minimize the sum of convex, quadratic cost functions depending on a common decision variable. By leveraging a recent system-theoretical reinterpretation of the considered algorithmic framework as a closed-loop linear dynamical system, the proposed approach generalizes the diagonal gain structure associated to the existing gradient tracking algorithms. Specifically, we look for closed-loop gain matrices that satisfy the sparsity constraints imposed by the network topology, without however being necessarily diagonal, as in existing gradient tracking schemes. We propose a novel procedure to compute stabilizing sparse gain matrices by solving a set of nonlinear matrix inequalities, based on the solution of a sequence of approximate linear versions of such inequalities. Numerical simulations are presented showing the enhanced performance of the proposed design compared to existing gradient tracking algorithms.

In this paper we consider a network of spatially distributed sensors which collect measurement samples of a spatial field, and aim at estimating in a distributed way (without any central coordinator) the entire field by suitably fusing all network data. We propose a general probabilistic model that can handle both partial knowledge of the physics generating the spatial field as well as a purely data-driven inference. Specifically, we adopt an Empirical Bayes approach in which the spatial field is modeled as a Gaussian Process, whose mean function is described by means of parametrized equations. We characterize the Empirical Bayes estimator when nodes are heterogeneous, i.e., perform a different number of measurements. Moreover, by exploiting the sparsity of both the covariance and the (parametrized) mean function of the Gaussian Process, we are able to design a distributed spatial field estimator. We corroborate the theoretical results with two numerical simulations: a stationary temperature field estimation in which the field is described by a partial differential (heat) equation, and a data driven inference in which the mean is parametrized by a cubic spline.