A note on the hybrid steepest descent methods

TL;DR

This paper establishes a convergence relationship between hybrid steepest descent and Halpern type iterative methods, showing that convergence of one implies convergence of the other under certain conditions.

Contribution

It provides a theoretical link demonstrating that hybrid steepest descent methods converge whenever Halpern type methods do, under specific settings.

Findings

Convergence of hybrid steepest descent methods is guaranteed if Halpern type methods converge.

The paper offers a theoretical framework connecting two iterative approaches.

Results apply under an appropriate mathematical setting.

Abstract

The aim of this paper is to prove that, in an appropriate setting, every iterative sequence generated by the hybrid steepest descent method is convergent whenever so is every iterative sequence generated by the Halpern type iterative method.