Distributed Asynchronous Projection onto the Intersection of Convex Sets

In this paper, we consider a distributed computation setting where agents are interconnected by a directed and strongly connected graph and the opinion of each agent is characterized by a convex set in a metric space. Given an initial point and under the assumption that the convex sets have a nonempty intersection, the aim of the agents is to compute the orthogonal projection onto the intersection of the convex sets. However, we assume each agent is able to compute the projection onto just its own convex set. Specifically, we consider an asynchronous token passing approach that extends traditional Dysktra's algorithm to the case where iterated projections onto the single convex sets are constrained by a network topology, considering two different token transmission policies: round-robin (e.g., a deterministic, cyclic ordering) and random (e.g., according to a Markov chain). The main strength points of the algorithm are the small memory and bandwidth requirements as well as the simplicity of implementation.