optimization and control

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

arXiv:2603.17105

summary

The paper uses proof‑mining techniques to derive explicit quantitative bounds on the asymptotic and T‑asymptotic regularity of an inexact generalized Halpern iteration, a viscosity‑type fixed‑point algorithm, and applies these results to related iterations such as the Kanzow‑Shehu method and the sequential averaging method.

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.

Topics & keywords

#proof mining#asymptotic regularity#halpern iteration#fixed point algorithms#viscosity methods#convergence analysisquantitative ratest-asymptotic regularityinexact iterationkanzow-shehu iterationsequential averaging methodlinear convergence