Strong convergence, perturbation resilience and superiorization of Generalized Modular String-Averaging with infinitely many input operators

TL;DR

This paper investigates the strong convergence and robustness of the Generalized Modular String-Averaging algorithm for infinite input operators, extending its applications to superiorization and dynamic string-averaging in Hilbert spaces.

Contribution

It introduces a unified framework for strong convergence and perturbation resilience of GMSA algorithms with infinitely many operators, including new algorithmic schemes.

Findings

Proves strong convergence for GMSA with infinite operators.

Establishes bounded perturbation resilience of the algorithms.

Demonstrates applicability to superiorization and dynamic string-averaging.

Abstract

We study the strong convergence and bounded perturbation resilience of iterative algorithms based on the Generalized Modular String-Averaging (GMSA) procedure for infinite sequences of input operators under a general admissible control. These methods address a variety of feasibility-seeking problems in real Hilbert spaces, including the common fixed point problem and the convex feasibility problem. In addition to the general case, involving certain strongly quasi-nonexpansive input operators, we consider a specific subclass of their corresponding relaxed firmly nonexpansive operators. This subclass proves useful for establishing bounded perturbation resilience. We further demonstrate the applicability of our strong convergence results, within the GMSA framework, to the Superiorization Methodology and to Dynamic String-Averaging, analyzing the behavior of a superiorized version of our main algorithm. The novelty and significance of this work is that it not only includes a variety of earlier algorithms as special cases but, more importantly, it allows the use of modular options of string-averaging that give rise to new, hitherto unavailable, algorithmic schemes with emphasis on infinitely many input operators. The strong convergence guarantees and the applications for superiorization and dynamic string-averaging are also important facets.