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Laurentiu Leustean

Publications and source records attributed to Laurentiu Leustean.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative asymptotic regularity and $T$-asymptotic regularity for the inexact generalized Halpern iteration

We apply proof mining techniques to obtain quantitative and qualitative results on asymptotic and T-asymptotic regularity for the inexact generalized Halpern iteration, a viscosity-type extension of an iteration recently studied by Kanzow and Shehu. Specializing our results to the Kanzow-Shehu iteration and the sequential averaging method (SAM) yields analogous results for these iterations. Furthermore, we compute rates of (T-)asymptotic regularity for particular choices of the parameter sequences, and for one of them, we obtain linear rates as an application of a lemma due to Sabach and Shtern.

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arXiv2025

Quantitative results on a generalized viscosity approximation method

In this paper, we study, in a nonlinear setting, the asymptotic behaviour of a generalized viscosity approximation method associated with a countable family of nonexpansive mappings satisfying resolvent-like conditions. We apply proof mining methods to obtain quantitative results on asymptotic regularity in W-hyperbolic spaces and rates of metastability in CAT(0) spaces.

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arXiv2023

On modified Halpern and Tikhonov-Mann iterations

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.

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arXiv2023

Rates of asymptotic regularity for the alternating Halpern-Mann iteration

In this paper we extend to $UCW$-hyperbolic spaces the quantitative asymptotic regularity results for the alternating Halpern-Mann iteration obtained by Dinis and the second author for CAT(0) spaces. These results are new even for uniformly convex normed spaces. Furthermore, for a particular choice of the parameter sequences, we compute linear rates of asymptotic regularity in $W$-hyperbolic spaces and quadratic rates of $T$- and $U$-asymptotic regularity in CAT(0) spaces.

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arXiv2022

Quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration

In this paper, we compute quadratic rates of asymptotic regularity for the Tikhonov-Mann iteration in W-hyperbolic spaces. This iteration is an extension to a nonlinear setting of the modified Mann iteration defined recently by Bot, Csetnek and Meier in Hilbert spaces. Furthermore, we show that the Douglas-Rachfors and forward-backward algorithms with Tikhonov regularization terms are special cases, in Hilbert spaces, of our Tikhonov-Mann iteration.

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arXiv2021

Quantitative results on a Halpern-type proximal point algorithm

We apply proof mining methods to analyse a result of Boikanyo and Moroşanu on the strong convergence of a Halpern-type proximal point algorithm. As a consequence, we obtain quantitative versions of this result, providing uniform effective rates of asymptotic regularity and metastability.

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arXiv2019

On the removal of weak compactness arguments in proof mining

The main observation of this paper is that some sequential weak compactness arguments in Hilbert space theory can be replaced by Heine/Borel compactness arguments (for the strong topology). Even though the latter form of compactness fails in (infinite-dimensional) Hilbert spaces, it nevertheless trivializes under the so-called bounded functional interpretation. As a consequence, the proof mining programme of extracting computational bounds from ordinary proofs of mathematics can be applied to {\em modified proofs} which use these false Heine/Borel compactness arguments. Additionally, the bounded functional interpretation provides good logical guidance in formulating quantitative versions of analytical statements. We illustrate these claims with three minings. The bounded functional interpretation is here used for the first time in proof mining.

math.LO↗
arXiv2018

An abstract proximal point algorithm

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.

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arXiv2018

Effective strong convergence of the proximal point algorithm in CAT(0) spaces

We apply methods of proof mining to obtain uniform quantitative bounds on the strong convergence of the proximal point algorithm for finding minimizers of convex, lower semicontinuous proper functions in CAT(0) spaces. Thus, for uniformly convex functions we compute rates of convergence, while, for totally bounded CAT(0) spaces we apply methods introduced by Kohlenbach, the first author and Nicolae to compute rates of metastability.

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arXiv2017

An application of proof mining to the proximal point algorithm in CAT(0) spaces

We compute, using techniques originally introduced by Kohlenbach, the first author and Nicolae, uniform rates of metastability for the proximal point algorithm in the context of CAT(0) spaces (as first considered by Bacak), specifically for the case where the ambient space is totally bounded. This result is part of the program of proof mining, which aims to apply methods of mathematical logic with the purpose of extracting quantitative information out of ordinary mathematical proofs, which may not be necessarily constructive.

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arXiv2016

Quantitative results on the Ishikawa iteration of Lipschitz pseudo-contractions

We compute uniform rates of metastability for the Ishikawa iteration of a Lipschitz pseudo-contractive self-mapping of a compact convex subset of a Hilbert space. This extraction is an instance of the proof mining program that aims to apply tools from mathematical logic in order to extract the hidden quantitative content of mathematical proofs. We prove our main result by applying methods developed by Kohlenbach, the first author and Nicolae for obtaining quantitative versions of strong convergence results for generalized Fejér monotone sequences in compact subsets of metric spaces.

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arXiv2015

Quantitative results on Fejer monotone sequences

We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejer monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many of the results hold in a general metric setting with some convexity structure added (so-called W-hyperbolic spaces). Sometimes uniform convexity is assumed still covering the important class of CAT(0)-spaces due to Gromov.

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arXiv2015

Effective results on nonlinear ergodic averages in CAT$(κ)$ spaces

In this paper we apply proof mining techniques to compute, in the setting of CAT$(κ)$ spaces (with $κ>0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.

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