Regular Sequences of Quasi-Nonexpansive Operators and Their Applications

TL;DR

This paper systematically studies regular sequences of quasi-nonexpansive operators in Hilbert space, demonstrating how their regularity properties influence convergence of iterative methods and applying these results to variational inequality problems.

Contribution

It establishes the preservation of regularity types under various operator operations and links regularity to convergence behavior in iterative algorithms.

Findings

Regularity types are preserved under relaxations, convex combinations, and products.

Weak, bounded, and linear regularity lead to weak, strong, and linear convergence.

Application to variational inequality problems demonstrates practical relevance.

Abstract

In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to block iterative and string averaging projection methods, which, in principle, are based on the above-mentioned algebraic operations applied to projections. Finally, we show an application of regular sequences of operators to variational inequality problems.