The Adaptive Projected Subgradient Method over the Fixed Point Set of Strongly Attracting Nonexpansive Mappings

Abstract

This paper presents an algorithmic solution, the adaptive projected subgradient method, to the problem of asymptotically minimizing a certain sequence of non-negative continuous convex functions over the fixed point set of a strongly attracting nonexpansive mapping in a real Hilbert space. The method generalizes Polyak's subgradient algorithm for the convexly constrained minimization of a fixed nonsmooth function. By generating a strongly convergent and asymptotically optimal point sequence, the proposed method not only offers unifying principles for many projection-based adaptive filtering algorithms but also enhances the adaptive filtering methods with the set theoretic estimation's armory by allowing a variety of a priori information on the estimandum in the form, for example, of multiple intersecting closed convex sets.