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.
Citation
Accepted for publication in Numerical Algorithms