Theory and applications of set theoretic adaptive filtering with multiple a-priori convex constraints - Part II: Proof of convergence theorem

摘要

Recently, the Adaptive Projected Subgradient Method (APSM) over multiple closed convex constraints has been proposed in order to tackle the problem of asymptotically minimizing a sequence of continuous, non-negative, and convex functions over multiple closed convex sets [Slavakis & Yamada, 2005 (Technical Report of IEICE-SIP, Jan. 2005)]. In this paper, by the fact that points satisfying multiple closed convex constraints can be seen as the fixed point set of strongly attracting nonexpansive mappings in a real Hilbert space, we provide with the proofs regarding the convergence theorem of the APSM over the fixed point set of strongly attracting nonexpansive mappings. In this way, these rigorous results firmly support the excellent performance of the APSM to various adaptive signal processing applications with multiple a-priori convex constraints like stereo echo cancelling and robust adaptive beamforming.