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An Approximate Projected Consensus Algorithm for Computing Intersection of Convex Sets

In this paper, we propose an approximate projected consensus algorithm for a network to cooperatively compute the intersection of convex sets. Instead of assuming the exact convex projection proposed in the literature, we allow each node to compute an approximate projection and communicate it to its neighbors. The communication graph is directed and time-varying. Nodes update their states by weighted averaging. Projection accuracy conditions are presented for the considered algorithm. They indicate how much projection accuracy is required to ensure global consensus to a point in the intersection set when the communication graph is uniformly jointly strongly connected. We show that $π/4$ is a critical angle error of the projection approximation to ensure a bounded state. A numerical example indicates that this approximate projected consensus algorithm may achieve better performance than the exact projected consensus algorithm in some cases.