Adaptive projected subgradient method and set theoretic adaptive filtering with multiple convex constraints

Abstract

This paper presents an algorithmic solution, the adaptive projected subgradient method, to the problem of asymptotically minimizing a certain sequence of nonnegative continuous convex functions over the fixed point set of strongly attracting nonexpansive mappings in a real Hilbert space. The proposed method provides with a strongly convergent, asymptotically optimal point sequence as well as with a characterization of the limiting point. As a side effect, the method allows the asymptotic minimization over the nonempty intersection of a finite number of closed convex sets. Thus, new directions for set theoretic adaptive filtering algorithms are revealed whenever the estimandum (system to be identified) is known to satisfy a number of convex constraints. This leads to a unification of a wide range of set theoretic adaptive filtering schemes such as NLMS, projected or constrained NLMS, APA, adaptive parallel subgradient projection algorithm, adaptive parallel min-max projection algorithm as well as their embedded constraint versions. Numerical results demonstrate the effectiveness of the proposed method to the problem of stereophonic acoustic echo cancellation.