Effective Rates of Convergence for the Resolvents of Accretive Operators

Koutsoukou-Argyraki, Angeliki

doi:10.1080/01630563.2017.1349146

Cited by 8 publications

(5 citation statements)

References 11 publications

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“…So far, proof mining has been successfully applied to obtain quantitative versions of celebrated results in various areas of mathematics such as approximation theory, nonlinear analysis, metric fixed point theory, ergodic theory, or topological dynamics. Recently, its methods have begun to be applied to convex optimization, for more details see [5,34,35,36,38,41,42].…”

Section: Introductionmentioning

confidence: 99%

An abstract proximal point algorithm

2018

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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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Section: Introductionmentioning

confidence: 99%

An abstract proximal point algorithm

2018

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“…Similar calculations along the lines of the above give rise to an instance of (⋆) where γ n represents some error between A n x n+1 and Ax n+1 . A series of quantitative results along these lines are given by Kohlenbach and the second author [34] and Koutsoukou-Argyraki [37], using a so-called modulus of uniform accretivity at zero defined by Kohlenbach and Koutsoukou-Argyraki [27] in place of the function ϕ.…”

Section: An Application For Projective Subgradient Methodsmentioning

confidence: 99%

A computational study of a class of recursive inequalities

Neri,

Powell

2023

J. Log. Anal.

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We examine the convergence properties of sequences of nonnegative real numbers that satisfy a particular class of recursive inequalities, from the perspective of proof theory and computability theory. We first establish a number of results concerning rates of convergence, setting out conditions under which computable rates are possible, and when not, providing corresponding rates of metastability. We then demonstrate how the aforementioned quantitative results can be applied to extract computational information from a range of proofs in nonlinear analysis. Here we provide both a new case study on subgradient algorithms, and give overviews of a selection of recent results which each involve an instance of our main recursive inequality. This paper contains the definitions of all relevant concepts from both proof theory and mathematical analysis, and as such, we hope that it is accessible to a general audience.

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“…The general version of the above lemma has been used in [24] to give rates of convergence for iterations associated to uniformly accretive operators in uniformly convex Banach spaces (see the end of this section), following earlier work that yielded rates of convergence for the gradient flow associated to such operators in [21], and for the asymptotic behaviour towards infinity of their resolvents and the fixed-step-size proximal point algorithm in [26]. The general notion underlying the strategy is that of a modulus of uniqueness, which was first introduced by Kohlenbach in the early 1990s [15,16,17] in the context of best approximation theory, while its significance in deriving rates of convergence for asymptotically regular iterative sequences was identified later in [23,Section 4.1].…”

Section: An Improvement On the Uniform Casementioning

confidence: 99%

Revisiting jointly firmly nonexpansive families of mappings

Sipoş¹

2020

Preprint

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Recently, the author, together with L. Leuştean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families.

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Effective Rates of Convergence for the Resolvents of Accretive Operators

Cited by 8 publications

References 11 publications

An abstract proximal point algorithm

An abstract proximal point algorithm

A computational study of a class of recursive inequalities

Revisiting jointly firmly nonexpansive families of mappings

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