A rate of metastability for the Halpern type Proximal Point Algorithm

TL;DR

This paper uses proof-theoretical methods to derive explicit rates of metastability and asymptotic regularity for the Halpern type proximal point algorithm, improving understanding of its convergence behavior.

Contribution

It provides the first primitive recursive quantitative rates for the algorithm's convergence, removing reliance on weak compactness arguments in the original proof.

Findings

Derived explicit rate of metastability for the algorithm

Obtained a rate of asymptotic regularity

Eliminated the need for sequential weak compactness in the proof

Abstract

Using proof-theoretical techniques, we analyze a proof by H.-K. Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of T. Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our final quantitative result bypasses the need of the sequential weak compactness argument present in the original proof. This elimination is reflected in the extraction of primitive recursive quantitative information. This work follows from recent results in Proof Mining regarding the removal of sequential weak compactness arguments.