On Modified Halpern and Tikhonov–Mann Iterations

Horaţiu Cheval (), Ulrich Kohlenbach () and Laurenţiu Leuştean ()
Additional contact information
Horaţiu Cheval: University of Bucharest
Ulrich Kohlenbach: Technische Universität Darmstadt
Laurenţiu Leuştean: University of Bucharest

Journal of Optimization Theory and Applications, 2023, vol. 197, issue 1, No 9, 233-251

Abstract: Abstract We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov–Mann iteration introduced by H. Cheval and L. Leuştean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Boţ et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Boţ et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov–Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leuştean and B. Dinis, P. Pinto, respectively. A transformation in the converse direction is also possible. We also obtain rates of asymptotic regularity of order O(1/n) for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov–Mann iteration in a general geodesic setting for a special choice of scalars.

Keywords: Mann iteration; Halpern iteration; Tikhonov regularization; Rates of asymptotic regularity; Rates of metastability; Proof mining; 47J25; 47H09; 03F10 (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-023-02192-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:197:y:2023:i:1:d:10.1007_s10957-023-02192-6

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-023-02192-6

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().