Constraint-Coupled Distributed Optimization: A Relaxation and Duality Approach

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

Distributed Mixed-Integer Linear Programming via Cut Generation and Constraint Exchange

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

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

ChoiRbot: A ROS 2 Toolbox for Cooperative Robotics

In this letter, 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

Convergence rate analysis of a subgradient averaging algorithm for distributed optimisation with different constraint sets

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) = rac{eta }{{k + 1}}, for some \u3b7 > 0. We then restrict attention to step-size choices c(k) = rac{eta }{{sqrt {k + 1} }},eta > 0, and establish a convergence of mathcal{O}left( {rac{{ln (k)}}{{sqrt k }}} ight) 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

Enhanced gradient tracking algorithms for distributed quadratic optimization via sparse gain design

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

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

Distributed nn-player approachability and consensus in coalitional games

We study a distributed allocation process where, repeatedly in time, every player renegotiates past allocations with neighbors and allocates new revenues. 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 make the grand coalitions stable

Network abstract linear programming with application to minimum-time formation control

We identify a novel class of distributed optimization problems, namely a networked version of abstract linear programming. For such problems we propose distributed algorithms for networks with various connectivity and/or memory constraints. Finally, we show how various minimum-time formation control problems can be tackled through appropriate geometric examples of abstract linear programs

Characterization of lipids preserved in Roman cooking pots by gas chromatography - mass spectrometry (GC-MS)

-EnLipid extracts of sherds of Roman cooking pots from the Temple of Apollo in Hierapolis (southwest Turkey) were analyzed using gas chromatography coupled with mass spectrometry (GCMS). The high relative abundance of saturated fatty acids, especially stearic acid, together with the presence of cholesterol, suggests that animal fats were processed in these pots. The detection of branched and odd chain fatty acids further indicate a possible ruminant source for these lipids. The data show also the presence of plant-derived products, like campesterol and b-sitosterol. The aim of this research is to illustrate how lipid analysis of pottery vessels, together with the information from archaeological context, can contribute to a better understanding of food habits and ritual activities in past ancient societies