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In this paper, we consider a network of processors that want to cooperatively solve a large-scale, convex optimization problem. Each processor has knowledge of a local cost function that depends only on a local variable. The goal is to minimize the sum of the local costs, while making the variables satisfy both local constraints and a global coupling constraint. We propose a simple, fully distributed algorithm, that works in a random, time-varying communication model, where at each iteration multiple edges are randomly drawn from an underlying graph. The algorithm is interpreted as a primal decomposition scheme applied to an equivalent problem reformulation. Almost sure convergence to the optimal cost of the original problem is proven by resorting to approaches from block subgradient methods. Specifically, the communication structure is mapped to a block structure, where the blocks correspond to the graph edges and are randomly selected at each iteration. Moreover, an almost sure asymptotic primal recovery property, with no averaging mechanisms, is shown. A numerical example corroborates the theoretical analysis.

In this paper we deal with a network of agents seeking to solve in a distributed way Mixed-Integer Linear Programs (MILPs) with a coupling constraint (modeling a limited shared resource) and local constraints. MILPs are NP-hard problems and several challenges arise in a distributed framework, so that looking for suboptimal solutions is of interest. To achieve this goal, the presence of a linear coupling calls for tailored decomposition approaches. We propose a fully distributed algorithm based on a primal decomposition approach and a suitable tightening of the coupling constraints. Agents repeatedly update local allocation vectors, which converge to an optimal resource allocation of an approximate version of the original problem. Based on such allocation vectors, agents are able to (locally) compute a mixed-integer solution, which is guaranteed to be feasible after a sufficiently large time. Asymptotic and finite-time suboptimality bounds are established for the computed solution. Numerical simulations highlight the efficacy of the proposed methodology. * This result is part of a project that has received funding

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.