A Rate of Metastability for the Halpern Type Proximal Point Algorithm

Using proof-theoretical techniques, we analyze a proof by Hong-Kun Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of Terence Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our f...

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Bibliographic Details
Published in:Numerical functional analysis and optimization Vol. 42; no. 3; pp. 320 - 343
Main Author: Pinto, Pedro
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 17.02.2021
Taylor & Francis Ltd
Subjects:
Algorithms
Metastability
Metastable state
proof mining
proximal point algorithm
rates of asymptotic regularity
sequential weak compactness
ISSN:0163-0563, 1532-2467
Online Access:Get full text
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Summary:Using proof-theoretical techniques, we analyze a proof by Hong-Kun Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of Terence Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our final quantitative result bypasses 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.
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ISSN:0163-0563
1532-2467
DOI:10.1080/01630563.2021.1876726