Below you can see a list of past talks at the Proof Mining Seminar in this season. For forthcoming talks, see the main page.
Wednesday, March 19, 2025
Paulo Firmino (University of Lisbon)
Rates of (T-)asymptotic regularity of the generalized Krasnoselskii-Mann-type iteration
Abstract:
In this talk we present the results of a quantitative analysis of work by Zhang, Guo, Wang [4] regarding the generalized Krasnoselskii-Mann-type iteration. We used proof mining methods to compute rates of (T-)asymptotic regularity of the iteration in a uniformly convex normed space, subsequently obtaining corollaries regarding the well-known Krasnoselskii-Mann iteration and other particular cases. We obtain quadratic rates for special choices of the parameter sequences.
This talk reports on recent joint work with Laurențiu Leuștean.
References:
[1] P. Firmino, L. Leuștean. Rates of (T-)asymptotic regularity of the generalized Krasnoselskii-Mann-type iteration. arXiv:2501.09523 [math.OC], 2025.
[2] L. Leuștean. Nonexpansive iterations in uniformly convex W-hyperbolic spaces. In B. S. Mordukhovich, I. Shafrir, and A. Zaslavski, editors, Nonlinear Analysis and Optimization I: Nonlinear Analysis, volume 513 of Contemporary Mathematics, pages 193–209. American Mathematical Society, 2010.
[3] L. Leuștean. An application of proof mining to nonlinear iterations. Annals of Pure and Applied Logic, 165:1484–1500, 2014.
[4] C. Kanzow and Y. Shehu. Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces. Computational Optimization and Applications, 67:595–620, 2017.
[5] Y.-C. Zhang, K. Guo, and T. Wang. Generalized Krasnoselskii-Mann-Type iteration for nonexpansive mappings in Banach spaces. Journal of the Operations Research Society of China, 9:195–206, 2021.
Wednesday, January 15, 2025
Pedro Pinto (TU Darmstadt)
On the Halpern method with adaptive anchoring parameters
Abstract:
In this talk, I will present a proof mining analysis of a recent result by He, Xu, Dong, and Mei [1], which introduces an accelerated version of the Halpern iteration with adaptively chosen anchoring parameters. Our quantitative study extends the convergence result from its original framework in Hilbert spaces to the broader nonlinear setting of Hadamard spaces. This generalization is achieved through a combination of certain optimizations and the elimination of weak compactness arguments, enabling the extraction of simple rates of metastability and fast rates of asymptotic regularity.
This talk reports on recent joint work with Nicholas Pischke.
References:
[1] S. He, H.-K. Xu, Q.-L. Dong, and N. Mei. Convergence analysis of the Halpern iteration with adaptive anchoring parameters. Mathematics of Computation, 93:327–345, 2024.
[2] P. Pinto, and N. Pischke. On the Halpern method with adaptive anchoring parameters. Oberwolfach Preprint OWP-2024-11, 2024.
[3] U. Kohlenbach. Some logical metatheorems with applications in functional analysis. Transactions of the American Mathematical Society, 357(1):89–128, 2005.
[4] U. Kohlenbach. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer-Verlag Berlin Heidelberg, 2008.
[5] F. Ferreira, L. Leuștean, and P. Pinto. On the removal of weak compactness arguments in proof mining. Advances in Mathematics, 354, 2019. 106728.