Reaching optimal consensus for multi-agent systems based on approximate projection

In this paper, we propose an approximately projected consensus algorithm (APCA) for a network to cooperatively compute the intersection of several convex sets, each of which is known only to a particular node. Instead of assuming the exact convex projection, we allow each node to just compute an approximate projection. The communication graph is directed and time-varying, and nodes can only exchange information via averaging among local view. We present sufficient and/or necessary conditions for the APCA, which shows on how much projection accuracy is required to ensure a global consensus within the intersection set when the communication graphs is uniformly jointly strongly connected. We show that π/4 is a critical angle error in the projection approximation to ensure a bounded solution for iterative projections. A numerical example indicates that the APCA may achieve better performance than the exact projected consensus algorithm. The results add the understanding of the fundamentals of distributed convex intersection computation.