Global behavior of parallel projection method for certain nonconvex feasibility problems

Abstract

Finding a common point of multiple closed sets in a real Hilbert space has been an important task in a wide range of signal processing. In this paper, we study asymptotic properties of the parallel projection method (PPM) for closed sets satisfying a special feasibility condition, which holds in the context of certain sparse signal processing. Our analysis guarantees that the cluster point set of PPM is exactly the intersection of the closed sets, and the distance to each set along a sequence generated by PPM with arbitrary initial point converges to zero. Moreover, under certain additional assumptions, we prove that the sequence converges to a point in the intersection of the closed sets, while existing analyses gave only local behaviors of PPM.