General Perturbation Resilient Dynamic String-Averaging for Inconsistent Problems with Superiorization

TL;DR

This paper introduces a new General Dynamic String-Averaging (GDSA) iterative scheme that converges in inconsistent problems and enhances the superiorization methodology, extending previous methods with stronger convergence guarantees.

Contribution

It develops the GDSA method combining dynamic string-averaging with strong coherence, providing convergence analysis for inconsistent problems and applications to superiorization.

Findings

Proves weak and strong convergence of GDSA in inconsistent cases.

Demonstrates bounded perturbation resilience of the GDSA method.

Extends superiorization methodology with new convergence results.

Abstract

In this paper we introduce a General Dynamic String-Averaging (GDSA) iterative scheme and investigate its convergence properties in the inconsistent case, that is, when the input operators don't have a common fixed point. The Dynamic String-Averaging Projection (DSAP) algorithm itself was introduced in an 2013 paper, where its strong convergence and bounded perturbation resilience were studied in the consistent case (that is, when the sets under consideration had a nonempty intersection). Results involving combination of the DSAP method with superiorization, were presented in 2015. The proof of the weak convergence of our GDSA method is based on the notion of "strong coherence" of sequences of operators that was introduced in 2019. This is an improvement of the property of "coherence" of sequences of operators introduced in 2001 by Bauschke and Combettes. Strong coherence provides a more convenient sufficient convergence condition for methods that employ infinite sequences of operators and it turns out to be a useful general tool when applied to proving the convergence of many iterative methods. In this paper we combine the ideas of both dynamic string-averaging and strong coherence, in order to analyze our GDSA method for a general class of operators and its bounded perturbation resilience in the inconsistent case with weak and strong convergence. We then discuss an application of the GDSA method to the Superiorization Methodology, developing results on the behavior of its superiorized version.