Abstract strongly convergent variants of the proximal point algorithm

Sipos, Andrei

doi:10.48550/arxiv.2108.13994

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2021

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“…The quantitative analysis allowed to generalize this reduction to the non-linear setting of hyperbolic spaces, which in turn allowed for several applications of previously known rates of metastability for these algorithms. Our final example can be found in [44] where the author adapts the analysis of the strong convergence of the Halpern type Proximal Point Algorithm, given in [28], from the setting of Banach spaces to CATp0q-spaces. This gave rise to new qualitative convergence results.…”

Section: Final Remarksmentioning

confidence: 99%

Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces

Dinis¹,

Pintó²

2021

Preprint

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In this paper we consider, in the general context of CATp0q spaces, an iterative schema which alternates between Halpern and Krasnoselskii-Mann style iterations. We prove, under suitable conditions, the strong convergence of this algorithm, benefiting from ideas from the proof mining program. We give quantitative information in the form of effective rates of asymptotic regularity and of metastability (in the sense of Tao). Motivated by these results we are also able to obtain strongly convergent versions of the forward-backward and the Douglas-Rachford algorithms. Our results generalize recent work by Bot ¸, Csetnek and Meier, and Cheval and Leuştean.

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Section: Final Remarksmentioning

confidence: 99%