A Simple Sufficient Condition for Convergence of Projected Consensus Algorithm

This letter studies the convergence property of the projected consensus algorithm. Under the assumption that: 1) the intersection of all constraint sets is nonempty; 2) the directed graph representing the communication among agents is time-invariant, strongly connected and aperiodic; and 3) the sum of the weights on incoming edges to each vertex of the graph is one, we prove that the states of all agents converge to a common point in the intersection of all constraint sets. Our proof does not need the assumption that every vertex has a self-loop. The validity of the theoretical analysis is confirmed through numerical experiments performed for a system of linear equations with nonnegativity constraints.