On the proximal point algorithm and its Halpern-type variant for generalized monotone operators in Hilbert space

In a recent paper, Bauschke et al. study ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comono...

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Bibliographic Details
Published in:Optimization letters Vol. 16; no. 2; pp. 611 - 621
Main Author: Kohlenbach, Ulrich
Format: Journal Article
Language:English
Published: Berlin, Heidelberg Springer 01.03.2022
Springer Berlin Heidelberg
Subjects:
Computational Intelligence
Generalized monotone operators
Halpern-type proximal point algorithm
Mathematics
Mathematics and Statistics
Metastability
Numerical and Computational Physics
Operations Research/Decision Theory
Optimization
Original Paper
Proof mining
Proximal point algorithm
Rates of convergence
Simulation
Halpern-type proximal point algorithm
Generalized monotone operators
Proximal point algorithm
Rates of convergence
Proof mining
Metastability
ISSN:1862-4480, 1862-4472, 1862-4480
Online Access:Get full text
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Summary:In a recent paper, Bauschke et al. study ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent JA.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_A.$$\end{document} In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. zerA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$zer\, A$$\end{document} we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}-comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.
ISSN:1862-4480
1862-4472
1862-4480
DOI:10.1007/s11590-021-01738-9