Random convex programs for distributed multi-agent consensus

We consider convex optimization problems with N randomly drawn convex constraints. Previous work has shown that the tails of the distribution of the probability that the optimal solution subject to these constraints will violate the next random constraint, can be bounded by a binomial distribution. In this paper we extend these results to the violation probability of convex combinations of optimal solutions of optimization problems with random constraints and different cost objectives. This extension has interesting applications to distributed multi-agent consensus algorithms in which the decision vectors of the agents are subject to random constraints and the agents' goal is to achieve consensus on a common value of the decision vector that satisfies the constraints. We give explicit bounds on the tails of the probability that the agents' decision vectors at an arbitrary iteration of the consensus protocol violate further constraint realizations. In a numerical experiment we apply these results to a model predictive control problem in which the agents aim to achieve consensus on a control sequence subject to random terminal constraints

Distributed Random Convex Programming via Constraints Consensus

This paper discusses distributed approaches for the solution of random convex programs (RCP). RCPs are convex optimization problems with a (usually large) number N of randomly extracted constraints; they arise in several applicative areas, especially in the context of decision under uncertainty, see [2],[3]. We here consider a setup in which instances of the random constraints (the scenario) are not held by a single centralized processing unit, but are distributed among different nodes of a network. Each node "sees" only a small subset of the constraints, and may communicate with neighbors. The objective is to make all nodes converge to the same solution as the centralized RCP problem. To this end, we develop two distributed algorithms that are variants of the constraints consensus algorithm [4],[5]: the active constraints consensus (ACC) algorithm, and the vertex constraints consensus (VCC) algorithm. We show that the ACC algorithm computes the overall optimal solution in finite time, and with almost surely bounded communication at each iteration. The VCC algorithm is instead tailored for the special case in which the constraint functions are convex also w.r.t. the uncertain parameters, and it computes the solution in a number of iterations bounded by the diameter of the communication graph. We further devise a variant of the VCC algorithm, namely quantized vertex constraints consensus (qVCC), to cope with the case in which communication bandwidth among processors is bounded. We discuss several applications of the proposed distributed techniques, including estimation, classification, and random model predictive control, and we present a numerical analysis of the performance of the proposed methods. As a complementary numerical result, we show that the parallel computation of the scenario solution using ACC algorithm significantly outperforms its centralized equivalent

A Distributed Algorithm for Random Convex Programming

International audienceWe study a distributed approach for solving random convex programs (RCP) for the case in which problem constraints are distributed among nodes in a processor network. We devise a distributed algorithm that allows network nodes to reach consensus on problem solution by exchanging a local set of constraints at each iteration. We prove that the algorithm assures finite-time convergence to problem solution and we provide explicit bounds on the maximum number of constraints to be exchanged among nodes at each communication round. Numerical experiments confirm the theoretical derivation and show that a parallel implementation of the proposed approach speeds-up the solution of the RCP with respect to centralized computation

A Distributed Algorithm for Random Convex Programming

International audienceWe study a distributed approach for solving random convex programs (RCP) for the case in which problem constraints are distributed among nodes in a processor network. We devise a distributed algorithm that allows network nodes to reach consensus on problem solution by exchanging a local set of constraints at each iteration. We prove that the algorithm assures finite-time convergence to problem solution and we provide explicit bounds on the maximum number of constraints to be exchanged among nodes at each communication round. Numerical experiments confirm the theoretical derivation and show that a parallel implementation of the proposed approach speeds-up the solution of the RCP with respect to centralized computation

A Distributed Algorithm for Random Convex Programming

We study a distributed approach for solving random convex programs (RCP) for the case in which problem constraints are distributed among nodes in a processor network. We devise a distributed algorithm that allows network nodes to reach consensus on problem solution by exchanging a local set of constraints at each iteration. We prove that the algorithm assures finite-time convergence to problem solution and we provide explicit bounds on the maximum number of constraints to be exchanged among nodes at each communication round. Numerical experiments confirm the theoretical derivation and show that a parallel implementation of the proposed approach speeds-up the solution of the RCP with respect to centralized computatio