arXiv subjects

Giuseppe Notarstefano

Publications and source records attributed to Giuseppe Notarstefano.

At least 55 records · Page 3Linked to original sources

arXiv2022

Uniform non-convex optimisation via Extremum Seeking

The paper deals with a well-known extremum seeking scheme by proving uniformity properties with respect to the amplitudes of the dither signal and of the cost function. Those properties are then used to show that the scheme guarantees the global minimiser to be semi-global practically stable despite the presence of local minima. Under the assumption of a globally Lipschitz cost function, it is shown that the scheme, improved through a high-pass filter, makes the global minimiser practically stable with a global domain of attraction.

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arXiv2022

Uniform quasi-convex optimisation via Extremum Seeking

The paper deals with a well-known extremum seeking scheme by proving uniformity properties with respect to the amplitudes of the dither signal and of the cost function. Those properties are then used to show that the scheme guarantees the global minimiser to be semi-global practically stable despite the presence of local saddle points. To achieve these results, we analyse the average system associated with the extremum seeking scheme via arguments based on the Fourier series.

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arXiv2021

Generalized Assignment for Multi-Robot Systems via Distributed Branch-And-Price

In this paper, we consider a network of agents that has to self-assign a set of tasks while respecting resource constraints. One possible formulation is the Generalized Assignment Problem, where the goal is to find a maximum payoff while satisfying capability constraints. We propose a purely distributed branch-and-price algorithm to solve this problem in a cooperative fashion. Inspired by classical (centralized) branch-and-price schemes, in the proposed algorithm each agent locally solves small linear programs, generates columns by solving simple knapsack problems, and communicates to its neighbors a fixed number of basic columns. We prove finite-time convergence of the algorithm to an optimal solution of the problem. Then, we apply the proposed scheme to a generalized assignment scenario in which a team of robots has to serve a set of tasks. We implement the proposed algorithm in a ROS testbed and provide experiments for a team of heterogeneous robots solving the assignment problem.

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arXiv2021

A Distributed Mixed-Integer Framework to Stochastic Optimal Microgrid Control

This paper deals with distributed control of microgrids composed of storages, generators, renewable energy sources, critical and controllable loads. We consider a stochastic formulation of the optimal control problem associated to 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 trained an open-source historical dataset of the EU.

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arXiv2021

Distributed Personalized Gradient Tracking with Convex Parametric Models

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.

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arXiv2021

Constraint-coupled Optimization with Unknown Costs: A Distributed Primal Decomposition Approach

In this paper, we present a distributed algorithm for solving convex, constraint-coupled, optimization problems over peer-to-peer networks. We consider a network of processors that aim to cooperatively minimize the sum of local cost functions, subject to individual constraints and to global coupling constraints. The major assumption of this work is that the cost functions are unknown and must be learned online. We propose a fully distributed algorithm, based on a primal decomposition approach, that uses iteratively refined data-driven estimations of the cost functions over the iterations. The algorithm is scalable and maintains private information of agents. We prove that, asymptotically, the distributed algorithm provides the optimal solution of the problem even though the true cost functions are never used within the algorithm. The analysis requires an in-depth exploration of the primal decomposition approach and shows that the distributed algorithm can be thought of as an epsilon-subgradient method applied to a suitable reformulation of the original problem. Finally, numerical computations cor- roborate the theoretical findings and show the efficacy of the proposed approach.

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arXiv2021

ChoiRbot: A ROS 2 Toolbox for Cooperative Robotics

In this paper, we introduce ChoiRbot, a toolbox for distributed cooperative robotics based on the novel Robot Operating System (ROS) 2. ChoiRbot provides a fully-functional toolset to execute complex distributed multi-robot tasks, either in simulation or experimentally, with a particular focus on networks of heterogeneous robots without a central coordinator. Thanks to its modular structure, ChoiRbot allows for a highly straight implementation of optimization-based distributed control schemes, such as distributed optimal control, model predictive control, task assignment, in which local computation and communication with neighboring robots are alternated. To this end, the toolbox provides functionalities for the solution of distributed optimization problems. The package can be also used to implement distributed feedback laws that do not need optimization features but do require the exchange of information among robots. The potential of the toolbox is illustrated with simulations and experiments on distributed robotics scenarios with mobile ground robots. The ChoiRbot toolbox is available at https://github.com/OPT4SMART/choirbot.

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arXiv2021

Distributed Primal Decomposition for Large-Scale MILPs

This paper deals with a distributed Mixed-Integer Linear Programming (MILP) set-up arising in several control applications. Agents of a network aim to minimize the sum of local linear cost functions subject to both individual constraints and a linear coupling constraint involving all the decision variables. A key, challenging feature of the considered set-up is that some components of the decision variables must assume integer values. The addressed MILPs are NP-hard, nonconvex and large-scale. Moreover, several additional challenges arise in a distributed framework due to the coupling constraint, so that feasible solutions with guaranteed suboptimality bounds are of interest. We propose a fully distributed algorithm based on a primal decomposition approach and an appropriate tightening of the coupling constraint. The algorithm is guaranteed to provide feasible solutions in finite time. Moreover, asymptotic and finite-time suboptimality bounds are established for the computed solution. Montecarlo simulations highlight the extremely low suboptimality bounds achieved by the algorithm.

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arXiv2020

Subgradient averaging for multi-agent optimisation with different constraint sets

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 $ \fracη{k+1} $, $ η> 0 $. We also analyse this scheme under a step size choice of $ \fracη{\sqrt{k+1}} $, $ η> 0 $, and establish a convergence rate of $ \mathcal{O}(\frac{\ln k}{\sqrt{k}}) $ in objective value. To demonstrate the efficacy of the proposed method, we investigate a robust regression problem and an $ \ell_2 $ regression problem with regularisation.

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arXiv2020

Distributed Constraint-Coupled Optimization via Primal Decomposition over Random Time-Varying Graphs

The paper addresses large-scale, convex optimization problems that need to be solved in a distributed way by agents communicating according to a random time-varying graph. Specifically, the goal of the network is to minimize the sum of local costs, while satisfying local and coupling constraints. Agents communicate according to a time-varying model in which edges of an underlying connected graph are active at each iteration with certain non-uniform probabilities. By relying on a primal decomposition scheme applied to an equivalent problem reformulation, we propose a novel distributed algorithm in which agents negotiate a local allocation of the total resource only with neighbors with active communication links. The algorithm is studied as a subgradient method with block-wise updates, in which blocks correspond to the graph edges that are active at each iteration. Thanks to this analysis approach, we show almost sure convergence to the optimal cost of the original problem and almost sure asymptotic primal recovery without resorting to averaging mechanisms typically employed in dual decomposition schemes. Explicit sublinear convergence rates are provided under the assumption of diminishing and constant step-sizes. Finally, an extensive numerical study on a plug-in electric vehicle charging problem corroborates the theoretical results.

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arXiv2020

Distributed Optimization for Smart Cyber-Physical Networks

The presence of embedded electronics and communication capabilities as well as sensing and control in smart devices has given rise to the novel concept of cyber-physical networks, in which agents aim at cooperatively solving complex tasks by local computation and communication. Numerous estimation, learning, decision and control tasks in smart networks involve the solution of large-scale, structured optimization problems in which network agents have only a partial knowledge of the whole problem. Distributed optimization aims at designing local computation and communication rules for the network processors allowing them to cooperatively solve the global optimization problem without relying on any central unit. The purpose of this survey is to provide an introduction to distributed optimization methodologies. Principal approaches, namely (primal) consensus-based, duality-based and constraint exchange methods, are formalized. An analysis of the basic schemes is supplied, and state-of-the-art extensions are reviewed.

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arXiv2020

Randomized Block Proximal Methods for Distributed Stochastic Big-Data Optimization

In this paper we introduce a class of novel distributed algorithms for solving stochastic big-data convex optimization problems over directed graphs. In the addressed set-up, the dimension of the decision variable can be extremely high and the objective function can be nonsmooth. The general algorithm consists of two main steps: a consensus step and an update on a single block of the optimization variable, which is then broadcast to neighbors. Three special instances of the proposed method, involving particular problem structures, are then presented. In the general case, the convergence of a dynamic consensus algorithm over random row stochastic matrices is shown. Then, the convergence of the proposed algorithm to the optimal cost is proven in expected value. Exact convergence is achieved when using diminishing (local) stepsizes, while approximate convergence is attained when constant stepsizes are employed. The convergence rate is shown to be sublinear and an explicit rate is provided in the case of constant stepsizes. Finally, the algorithm is tested on a distributed classification problem, first on synthetic data and, then, on a real, high-dimensional, text dataset.

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arXiv2020

Distributed Submodular Minimization via Block-Wise Updates and Communications

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 Lovàsz 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.

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arXiv2020

DISROPT: a Python Framework for Distributed Optimization

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.

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arXiv2020

On the Linear Convergence Rate of the Distributed Block Proximal Method

The recently developed Distributed Block Proximal Method, for solving stochastic big-data convex optimization problems, is studied in this paper under the assumption of constant stepsizes and strongly convex (possibly non-smooth) local objective functions. This class of problems arises in many learning and classification problems in which, for example, strongly-convex regularizing functions are included in the objective function, the decision variable is extremely high dimensional, and large datasets are employed. The algorithm produces local estimates by means of block-wise updates and communication among the agents. The expected distance from the (global) optimum, in terms of cost value, is shown to decay linearly to a constant value which is proportional to the selected local stepsizes. A numerical example involving a classification problem corroborates the theoretical results.

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arXiv2020

A Sparse Polytopic LPV Controller for Fully-Distributed Nonlinear Optimal Control

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.

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arXiv2019

A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

In this paper we consider a recently developed distributed optimization algorithm based on gradient tracking. We propose a system theory framework to analyze its structural properties on a preliminary, quadratic optimization set-up. Specifically, we focus on a scenario in which agents in a static network want to cooperatively minimize the sum of quadratic cost functions. We show that the gradient tracking distributed algorithm for the investigated program can be viewed as a sparse closed-loop linear system in which the dynamic state-feedback controller includes consensus matrices and optimization (stepsize) parameters. The closed-loop system turns out to be not completely reachable and asymptotic stability can be shown restricted to a proper invariant set. Convergence to the global minimum, in turn, can be obtained only by means of a proper initialization. The proposed system interpretation of the distributed algorithm provides also additional insights on other structural properties and possible design choices that are discussed in the last part of the paper as a starting point for future developments.

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arXiv2019

Tracking-ADMM for Distributed Constraint-Coupled Optimization

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.

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