Alternating projection method for intersection of convex sets, multi-agent consensus algorithms, and averaging inequalities

The history of the alternating projection method for finding a common point of several convex sets in Euclidean space goes back to the well-known Kaczmarz algorithm for solving systems of linear equations, which was devised in the 1930s and later found wide applications in image processing and computed tomography. An important role in the study of this method was played by I.I. Eremin’s, L.M. Bregman’s, and B.T. Polyak’s works, which appeared nearly simultaneously and contained general results concerning the convergence of alternating projections to a point in the intersection of sets, assuming that this intersection is nonempty. In this paper, we consider a modification of the convex set intersection problem that is related to the theory of multi-agent systems and is called the constrained consensus problem. Each convex set in this problem is associated with a certain agent and, generally speaking, is inaccessible to the other agents. A group of agents is interested in finding a common point of these sets, that is, a point satisfying all the constraints. Distributed analogues of the alternating projection method proposed for solving this problem lead to a rather complicated nonlinear system of equations, the convergence of which is usually proved using special Lyapunov functions. A brief survey of these methods is given, and their relation to the theorem ensuring consensus in a system of averaging inequalities recently proved by the second author is shown (this theorem develops convergence results for the usual method of iterative averaging as applied to the consensus problem).