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In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean square in a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.

In this paper we propose a distributed implementation
of the relaxed Alternating Direction Method of Multipliers algorithm
(R-ADMM) for optimization of a separable convex cost
function, whose terms are stored by a set of interacting agents,
one for each agent. Specifically  the local cost stored by each node is in
general a function of both the state of the node and the states of its
neighbors, a framework that we refer to as `partition-based' optimization.
This framework presents a great flexibility and can be adapted to a large
number of different applications.
By recasting the problem into an operator theoretical framework, the proposed
algorithm is shown to be provably robust against random packet losses that
might occur in the communication between
neighboring nodes. Finally, the effectiveness of the proposed algorithm is
confirmed by  a set of compelling numerical simulations run over random
geometric graphs subject to i.i.d. random packet losses.

In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMM-like algorithm, referred to as relaxed ADMM (R-ADMM) via a generalized Peaceman-Rachford Splitting operator on the Lagrange dual formulation of the original optimization problem. This algorithm depends on two parameters, namely the averaging coefficient $\alpha$ and the augmented Lagrangian coefficient $\rho$ and we show that by setting $\alpha=1/2$ we recover the standard ADMM algorithm as a special case. Moreover, first, we reformulate our R-ADMM algorithm into an implementation that presents reduced complexity in terms of memory, communication and computational requirements. Second, we propose a further reformulation which let us provide the first ADMM-like algorithm with guaranteed convergence properties even in the presence of lossy communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over random geometric graphs subject to i.i.d. random packet losses.

We consider the problem of controlling a smart lighting system of multiple luminaires with collocated occupancy and light sensors. The objective is to attain illumination levels higher than specified values (possibly changing over time) at the workplace by adapting dimming levels using sensor information, while minimizing energy consumption. We propose to estimate the daylight illuminance levels at the workplace based on the daylight illuminance measurements at the ceiling. More specifically, this daylight estimator is based on a model built from data collected by light sensors placed at workplace reference points and at the luminaires in a training phase. Three estimation methods are considered: Regularized least squares, locally weighted regularized least squares, and cluster-based regularized least squares. This model is then used in the operational phase by the lighting controller to compute dimming levels by solving a linear programming problem, in which power consumption is minimized under the constraint that the estimated illuminance is higher than a specified target value. The performance of the proposed approach with the three estimation methods is evaluated using an open-office lighting model with different daylight conditions. We show that the proposed approach offers reduced under-illumination and energy consumption in comparison to existing alternative approaches.

In this paper, we study the problem of real-time optimal distributed partitioning for perimeter patrolling in the context of multicamera networks for surveillance, where each camera has limited mobility range and speed, and the communication is unreliable. The objective is to coordinate the cameras in order to minimize the time elapsed between two different visits of each point of the perimeter. We address this problem by casting it into a convex problem in which the perimeter is partitioned into nonoverlapping segments, each patrolled by a camera that sweeps back and forth at the maximum speed. We then propose an asynchronous distributed algorithm that guarantees that these segments cover the whole patrolling perimeter at any time and asymptotically converge to the optimal centralized solution under reliable communication. We finally modify the proposed algorithm in order to attain the same convergence and covering properties even in the more challenging scenario, where communication is lossy and there is no channel feedback, i.e., the transmitting camera is not aware whether a packet has been received or not by its neighbors.

The problem of daylight estimation in a smart lighting system is considered. The smart lighting system consists of multiple luminaires with collocated occupancy and light sensors. Using sensor information, the objective is to attain illumination levels higher than specified values at the workspaces. We consider a training phase wherein light sensors are used at the workspaces in addition. Data from the light sensors at the ceiling and workspaces is used to estimate the mapping across the sensors. In the operational phase, the estimated mapping is used at the lighting controller to obtain an estimate of the illuminance value at the workspaces. Under the constraint that the estimated illuminance is higher than a specified target value, the controller optimizes the dimming levels of the luminaires to minimize power consumption. We evaluate the performance of the proposed approach in an open-office lighting model by considering different daylight conditions.

We address the problem of distributed unconstrained convex optimization under separability assumptions, i.e., the framework where a network of agents, each endowed with local private multidimensional convex cost and subject to communication constraints, wants to collaborate to compute the minimizer of the sum of the local costs. We propose a design methodology that combines average consensus algorithms and separation of time-scales ideas. This strategy is proven, under suitable hypotheses, to be globally convergent to the true minimizer. Intuitively, the procedure lets the agents distributedly compute and sequentially update an approximated Newton-Raphson direction by means of suitable average consensus ratios. We show with numerical simulations that the speed of convergence of this strategy is comparable with alternative optimization strategies such as the Alternating Direction Method of Multipliers. Finally, we propose some alternative strategies which trade-off communication and computational requirements with convergence speed.

In this paper we analyze the performances of some popular algorithms used to solve distributed optimization problems involving quadratic cost functions in a multi agent system. Namely, we study the performances of standard consensus, accelerated consensus and ADMM.  We analyze the scalar quadratic function case, under different scenarios and with structured graphs. We find that accelerated consensus is the algorithm with the best performance in all the cases analyzed. On the other hand, ADMM has performance comparable to the accelerated consensus when the graph is scarcely connected, while for dense graphs its performance deteriorates and becomes worse than the one of standard consensus. The results therefore suggest that the choice of the algorithm to solve the problem we analyze strongly depends on the graph, and that accelerated consensus should always be preferred.

Personal control with occupancy and daylight adaptation is considered in a lighting system with multiple luminaires. Each luminaire is equipped with a co-located occupancy sensor and light sensor that respectively provide local occupancy and illumination information to a central controller. Users may also provide control inputs to indicate a desired illuminance value. Using sensor feedback and user input, the central controller determines dimming values of the luminaires using an optimization framework. The cost function consists of a weighted sum of illumination errors at light sensors and the power consumption of the system. The optimum dimming values are determined with the constraints that the illuminance value at the light sensors are above the reference set-point at the light sensors and the dimming levels are within physical allowable limits. Different approaches to determine the set-points at light sensors associated with multiple user illumination requests are considered. The performance of the proposed constrained optimization problem is compared with a reference stand-alone controller under different simulation scenarios in an open-plan office lighting system.

The knowledge of the size of a network, i.e.\ of the number of nodes composing it, is important for maintenance and organization purposes. In networks where the identity of the nodes or is not unique or cannot be disclosed for privacy reasons, the size-estimation problem is particularly challenging since the exchanged messages cannot be uniquely associated with a specific node. In this work, we propose a totally distributed anonymous strategy based on statistical inference concepts. In our approach, each node starts generating a vector of independent random numbers from a known distribution. Then nodes compute a common function via some distributed consensus algorithms, and finally they compute the Maximum Likelihood (ML) estimate of the network size exploiting opportune statistical inferences. In this work we study the performance that can be obtained following this computational scheme when the consensus strategy is either the maximum or the average. In the max-consensus scenario, when data come from absolutely continuous distributions, we provide a complete characterization of the ML estimator. In particular, we show that the squared estimation error decreases as $1/M$, where $M$ is the amount of random numbers locally generated by each node, independently of the chosen probability distribution. Differently, in the average-consensus scenario, we show that if the locally generated data are independent Bernoulli trials, then the probability for the ML estimator to return a wrong answer decreases exponentially in $M$. Finally, we provide a discussion as how the numerical errors may affect the estimators performance under different scenarios.

This paper considers a variation of the 17$^{\text{th}}$ century problem commonly known as the Newton-Pepys problem, or the John Smith's problem. We provide its solution, interpreting the result in terms of maximum likelihood estimation and Ockham's razor. In addition, we illustrate the practical relevance of these findings for solving size-estimation problems, and in particular for determining the number of agents in a wireless sensor network.

The paper considers the classification problem using Support Vector Machines, and investigates how to maximally reduce the size of the training set without losing information. Under linearly separable dataset assumptions, we derive the exact conditions stating which observations can be discarded without diminishing the overall information content. For this purpose, we introduce the concept of Potential Support Vectors, i.e., those data that can become Support Vectors when future data become available. Complementary, we also characterize the set of Discardable Vectors, i.e., those data that, given the current dataset, can never become Support Vectors. These vectors are thus useless for future training purposes, and can eventually be removed without loss of information. We then provide an efficient algorithm based on linear programming which returns the potential and discardable vectors by constructing a simplex tableau. Finally we compare it with alternative algorithms available in the literature on some synthetic data as well as on datasets from standard repositories.

We consider the distributed unconstrained minimization of separable convex costfunctions, where the global cost is given by the sum of several local and private costs, eachassociated to a specific agent of a given communication network. We specifically address anasynchronous distributed optimization technique called Newton-Raphson consensus. Besidehaving low computational complexity, low communication requirements and being interpretableas a distributed Newton-Raphson algorithm, the technique has also the beneficial properties ofrequiring very little coordination and naturally support time-varying topologies. In this workwe analytically prove that under some assumptions it shows local convergence properties, andcorroborate this result by means of numerical simulations.

To maintain and organize distributed systems it is necessary to have a certain degree of knowledge of their status like the number of cooperating agents. The estimation of this number, usually referred as the network size, can pose challenging questions when agents' identification information cannot be disclosed, since the exchanged information cannot be associated to who originated it. In this paper we propose a totally distributed network size estimation strategy based on statistical inference concepts that can be applied under anonymity constraints. The scheme is based on the following paradigm: agents locally generate some Bernoulli trials, then distributedly compute averages of these generated data, finally locally compute the Maximum Likelihood estimate of the network size exploiting its probabilistic dependencies with the previously computed averages. In this work we study the statistical properties of this estimation strategy, and show how the probability of returning a wrong evaluation decreases exponentially in the number of locally generated trials. Finally, we discuss how practical implementation issues may affect the estimator, and show that there exists a neat phase transition between insensitivity to numerical errors and uselessness of the results.

In this paper we address the problem of exploitingthe distributed energy resources (DER) available in a smartmicro-grid to minimize the power distribution losses via optimalreactive power compensation. Due to their typically smallsize, the amount of reactive power provided by each micro-generator is subject to tight saturation constraints. As aconsequence, it might be impossible to achieve convergence tothe global optimum based on algorithms that rely on short-range, gossip-type communication. We therefore propose arandomized multi-hop protocol that guarantees convergence ofthe distributed optimization algorithm also when only short-range communications are possible, at the expense of someadditional communication overhead.

In this paper we focus on collaborative wireless sensor networks, where agents are randomly distributed over a region of interest and collaborate to achieve a common estimation goal. In particular, we introduce two consensus-based distributed linear estimators. The first one is designed for a Bayesian scenario, where an unknown common finite-dimensional parameter vector has to be reconstructed,  while the second one  regards the nonparametric reconstruction of an unknown function sampled at different locations by the sensors. Both of the algorithms are characterized in terms of the trade-off between estimation performance, communication, computation and memory complexity. In the finite-dimensional setting, we derive mild sufficient conditions which ensure that distributed estimator performs better than the local optimal ones in terms of estimation error variance. In the nonparametric setting, we introduce an on-line algorithm that allows the agents both to compute the function estimate with small computational, communication and data storage efforts, and to quantify its distance from the centralized estimate given by a Regularization Network, one of the most powerful regularized kernel methods. These results are obtained by deriving bounds on the estimation error that provide insights on how the uncertainty inherent in a sensor network, such as imperfect knowledge on the number of agents and the measurement models used by the sensors, can degrade the performance of the estimation process. Numerical experiments are included to support the theoretical findings.

In this paper we study the problem of real-time optimal distributed partitioning for perimeter patrolling in the context of multi-camera networks for surveillance. The objective is to partition a given segment into non-overlapping sub-segments, each assigned to a different camera to patrol. Each camera has both physical mobility range and limited speed, and it must patrol its assigned sub-segment by sweeping it back and forth at maximum speed. Here we first review the solution for the centralized optimal partitioning. Then we propose two different distributed control strategies to determine the extremes of the optimal patrolling areas of each camera. Both these strategies require only local communication with the neighboring cameras but adopt different communication schemes, respectively, symmetric gossip and asynchronous asymmetric broadcast. The first scheme is shown to be provably convergent to the optimal solution. Some theoretical insights are provided also for the second scheme whose effectiveness is validated through numerical simulations.

In this work we consider a multidimensional distributed optimization technique that is suitable for multiagents systems subject to limited communication connectivity. In particular, we consider a convex unconstrained additive problem, i.e. a case where the global convex unconstrained multidimensional cost function is given by the sum of local cost functions available only to the specific owning agents. We show how, by exploiting the separation of time-scales principle,the multidimensional consensus-based strategy approximates a Newton-Raphson descent algorithm. We propose two alternative optimization strategies corresponding to approximations of the main procedure. These approximations introduce tradeoffs between the required communication bandwidth and the convergence speed/accuracy of the results. We provide analytical proofs of convergence and numerical simulations supporting the intuitions developed through the paper.

We consider the convergence rates of two peculiar2 convex optimization strategies in the context of multi agent3 systems, namely the Newton-Raphson consensus optimization4 and a distributed Gradient-Descent opportunely derived from5 the first. To allow analytical derivations, the convergence6 analyses are performed under the simplificative assumption of7 quadratic local cost functions. In this framework we derive8 sufficient conditions which guarantee the convergence of the9 algorithms. From these conditions we then obtain closed form10 expressions that can be used to tune the parameters for11 maximizing the rate of convergence. Despite these formulae12 have been derived under quadratic local cost functions13 assumptions, they can be used as rules-of-thumb for tuning14 the parameters of the algorithms in general situations.

In this work we study the problem of unconstrained distributed optimization in the context of multi-agents systems subject to limited communication connectivity. In particular we focus on the minimization of a sum of convex cost functions, where each component of the global function is available only to a specific agent and can thus be seen as a private local cost. The agents need to cooperate to compute the minimizer of the sum of all costs. We propose a consensus-like strategy to estimate a Newton-Raphson descending update for the local estimates of the global minimizer at each agent. In particular, the algorithm is based on the separation of time-scales principle and it is proved to converge to the global minimizer if a specific parameter that tunes the rate of convergence is chosen sufficiently small. We also provide numerical simulations and compare them with alternative distributed optimization strategies like the Alternating Direction Method of Multipliers and the Distributed Subgradient Method.

We analyze the problem of data sets reduction

for support vector classification. The work is also motivated

by distributed problems, where sensors collect binary mea-

surements at different locations moving inside an environment

that needs to be divided into a collection of regions labeled in

two different ways. The scope is to let each agent retain and

exchange only those measurements that are mostly informative

for the collective reconstruction of the decision boundary. For

the case of separable classes, we provide the exact conditions

and an efficient algorithm to determine if an element in the

training set can become a support vector when new data arrive.

The analysis is then extended to the non-separable case deriving

a sufficient discardability condition and a general data selection

scheme for classification. Numerical experiments relative to the

distributed problem show that the proposed procedure allows

the agents to exchange a small amount of the collected data to

obtain a highly predictive decision boundary.

In this paper we study the problem of estimating the channel parameters for a generic wireless sensor network (WSN) in a completely distributed manner, using consensus algorithms. Specifically, we first propose a distributed strategy to minimize the effects of unknown constant offsets in the reading ofthe Radio Strength Signal Indicator (RSSI) due to uncalibrated sensors. Then we show how the computation of the optimal wireless channels parameters, which are the solution of a global least-square optimization problem, can be obtained with a consensus-based algorithm. The proposed algorithms are general algorithms for sensor calibration and distributed least-square parameter identification,and do not require any knowledge on the global topology of the network nor the total number of nodes. Finally, we apply these algorithms to experimental data collected from an indoor wireless sensor network.

The distributed estimation of the number of active sensors in a network can be important for estimation and organization purposes. We propose a design methodology based on the following paradigm: some locally randomly generated values are exchanged among the various sensors and thus modified by known consensus-based strategies. Statistical analysis of the a-consensus values allows estimation of the number of participant sensors. The main features of this approach are: algorithms are completely distributed, since they do not require leader election steps; sensors are not requested to transmit authenticative information (for example identificative numbers or similar data), and thus the strategy can be implemented whenever privacy problems arise. After a rigorous formulation of the paradigma we analyze some practical examples, fully characterize them from a statistical point of view, and finally we provide some general theoretical results and asymptotic analyses.

In this paper we analyze the problem of estimating a function from different noisy data sets collected by spatially distributed sensors and subject to unknown temporal shifts. We propose a novel approach based on non-parametric Gaussian regression and reproducing kernel Hilbert space theory that exploit compact and accurate representations of function estimates. As a first result, suitable minimization of inner products in reproducing kernel Hilbert spaces is used to obtain a novel time delay estimation technique when attention is restricted only to two signals. Then, we derive both a centralized and a distributed estimator to simultaneously identify the unknown function and delays for a generic number of networked sensors subject to a restricted communication graph. Numerical simulations are used to test the effectiveness of the proposed approaches.

In this paper we study the problem of estimatingthe channel parameters for a generic wireless sensor network(WSN) in a completely distributed manner, using consensusalgorithms. Specifically, we first propose a distributed strategyto minimize the effects of unknown constant offsets in thereading of the Radio Strength Signal Indicator (RSSI) due touncalibrated sensors. Then we show how the computation of theoptimal wireless channels parameters, which are the solutionof a global least-square optimization problem, can be obtainedwith a consensus-based algorithm. The proposed algorithmsare general algorithms for sensor calibration and distributedleast-square parameter identification, and do not require anyknowledge on the global topology of the network nor thetotal number of nodes. Finally, we apply these algorithms toexperimental data collected from an indoor WSN.