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

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.