On the convergence of algorithms with Tikhonov regularization terms

Dinis, Bruno; Pinto, Pedro

doi:10.1007/s11590-020-01635-7

Cited by 13 publications

(22 citation statements)

References 9 publications

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“…These quantitative results may also be seen as a natural continuation of previous quantitative analyses (see e.g. [13,30,12]). Even though our results and proofs are inspired by proof theoretical techniques, these are only used as an intermediate step and are not visible in the final product.…”

Section: Introductionsupporting

confidence: 68%

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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: Introductionsupporting

confidence: 68%

“…Nevertheless, the technique developed in [16] shows that, through a quantitative treatment, it is possible to bypass this argument in Hilbert spaces (as was done e.g. in [13,14]). Extending such quantitative results to the setting of CATp0q spaces allows to conclude the main theorem.…”

Section: Introductionmentioning

confidence: 99%

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

Dinis¹,

Pintó²

2021

Preprint

Self Cite

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“…The fact that (18) holds for all n ∈ N follows from (7). Furthermore, L is an upper bound on (s n ), as d(x n+1 , x n ) ≤ d(x n+1 , p) + d(x n , p) ≤ 2M , by Lemma 3.1.…”

Section: Proofs Of the Main Theoremsmentioning

confidence: 87%

“…A quantitative analysis of the proof of Theorem 1.1 was obtained recently by Dinis and Pinto (see [7,Lemma 5]).…”

Section: (Ii) (C3) (C4) (C5) and (C6)mentioning

confidence: 99%

Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration

Cheval¹,

Leuştean²

2021

Preprint

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“…As remarked before, results -including quantitative ones -concerning Tikhonov-regularized algorithms may be obtained from the corresponding Halpern ones, as per [33]; see also [14,Section 3.3] for examples which specifically concern metastability. (The study of Tikhonov-regularized algorithms was also studied from the viewpoint of proof mining in [13,12].) We may now, thus, state the corresponding quantitative version of Corollary 3.5.…”

Section: Thenmentioning

confidence: 99%