Por Pedro Pinto (Technische Universität Darmstadt).

In this talk I shall discuss recent joint work with Bruno Dinis, where we considered an iterative schema which alternates between Halpern and Krasnoselskii-Mann style iterations.
We prove, under suitable conditions, the strong convergence of this algorithm in the general nonlinear setting of CAT(0) spaces. Besides obtaining quantitative information, we will see how such proof was made possible by techniques and ideas from the proof mining program.
Our results generalize recent work by Boț, Csetnek and Meier, and Cheval and Leustean.

References:

1. B. Dinis and P. Pinto. Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces. arXiv:2112.14525, 2021.
2. R. I. Bot, E. R. Csetnek, and D. Meier. Inducing strong convergence into the asymptotic behaviour of proximal splitting algorithms in Hilbert spaces. Optim. Methods Softw., 34(3):489{514, 2019.
3. H. Cheval and L. Leuștean. Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration. arXiv:2107.07176, 2021.
4. F. Ferreira, L. Leuștean, and P. Pinto. On the removal of weak compactness arguments in proof mining. Advances in Mathematics, 354:106728, 2019
5. U. Kohlenbach and L. Leuștean. Effective metastability of Halpern iterates in CAT(0) spaces. Advances in Mathematics, 231(5):2526{2556, 2012.

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