This paper presents a robust, distributed algorithm to solve general linear programs. The algorithm design builds on the characterization of the solutions of the linear program as saddle points of a modified Lagrangian function. We show that the resulting continuous-time saddle-point algorithm is provably correct but, in general, not distributed because of a global parameter associated with the nonsmooth exact penalty function employed to encode the inequality constraints of the linear program. This motivates the design of a discontinuous saddle-point dynamics that, while enjoying the same convergence guarantees, is fully distributed and scalable with the dimension of the solution vector. We also characterize the robustness against disturbances and link failures of the proposed dynamics. Specifically, we show that it is integral-input-to-state stable but not input-to-state stable. The latter fact is a consequence of a more general result, that we also establish, which states that no algorithmic solution for linear programming is input-to-state stable when uncertainty in the problem data affects the dynamics as a disturbance. Our results allow us to establish the resilience of the proposed distributed dynamics to disturbances of finite variation and recurrently disconnected communication among the agents. Simulations in an optimal control application illustrate the results.

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We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.AMS subject classifications. 90C05, 68M14, 93C30, 65K10, 93C651. Introduction. The global objective of many multi-agent systems can be formulated as an optimization problem where the individual agents' states are the decision variables. Due to the inherent networked structure of these problems, much research has been devoted to developing local dynamics for each agent that guarantee that the aggregate of their states converge to a solution of the optimization problem. From an analysis viewpoint, the availability of powerful concepts and tools from stability analysis makes continuous-time coordination algorithms appealing. However, their implementation requires the continuous flow of information among agents. On the other hand, discrete-time algorithms are amenable to real-time implementation, but the selection of the stepsizes to guarantee convergence has to take into account worst-case situations, leading to an inefficient use of the network resources. In this paper, we seek to combine the advantages of both approaches by designing a distributed algorithmic solution to linear programming in standard form that combines continuous-time computation by individual agents with opportunistic event-triggered communication among neighbors. Our focus on linear programming is motivated by its importance in mathematical optimization and its pervasiveness in multi-agent scenarios, with applications to task assignment, network flow, optimal control, and energy storage, among others.Literature review. The present work has connections with three main areas: distributed optimization, event-triggered control, and switched and hybrid systems. Distributed convex optimization problems have many applications to networked systems, see e.g., [2,20,24], and this has motivated the development of a growing body of work that includes dual-decomposition [22,27], the alternating direction method of multipliers [26], subgradient projection algorithms [14,19,28], auction algorithms [1], and saddle-point dynamics [6,7]. The works [4,21] propose algor...

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Discussion during the forum revealed several benefits of creating ethical safe space. This model of workshop allows space for participation of stakeholders, who might not otherwise be able to interact in the same forum, to articulate their perspectives and debate with other presenters and audience members. Participants at the forum spoke of the creation of ethical safe space as a starting point for more dialogue on the issues raised by the policy statement. The forum was, therefore, seen as a potential starting point for building conversation that would facilitate revising the policy with broader consultation on its legal and ethical validity.