Quantitative asymptotic regularity and $T$-asymptotic regularity for the inexact generalized Halpern iteration

TL;DR

This paper uses proof mining to derive quantitative and qualitative results on the asymptotic regularity of the inexact generalized Halpern iteration, extending previous work and providing explicit rates of convergence.

Contribution

It introduces new quantitative bounds for the inexact generalized Halpern iteration and related methods, including linear rates in specific cases.

Findings

Derived explicit rates of (T-)asymptotic regularity.

Extended results to the Kanzow-Shehu iteration and SAM.

Obtained linear convergence rates for particular parameter choices.

Abstract

We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.