The Bregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces
type: Item
instance of: scholarly article
title: The Bregman distance, approximate compactness and convexity of Chebyshev sets in Banach spaces
zbMATH Open document ID: 1198.46012
author: C. Li
author: Wen Song
author: Jen-Chih Yao
published in: Journal of Approximation Theory
publication date: 2010-06-25T00:00:00Z
review text: Let \(X\) be a real Banach space and \(g:X\to\bar{\mathbb R}=\mathbb R\cup\{\infty\}\) be a proper convex function. The Bregman distance corresponding to \(g\) is defined by \(D_g(y,x)=g(y-g(x)-g'_+(x;y-x)\) for \(x,y\in\text{dom}\,g\), where \(g'_+(x;\cdot)\) stands for the right-hand directional derivative of \(g\) at \(x\). The paper is concerned with the famous problem of convexity of Chebyshev sets -- a subset \(C\subset \) int (dom \(g\)) is called \(D_g\)-Chebyshev if the \(D_g\)-projection \(\Pi^g_C(x)= \) arg min\(_{y\in C} D_g(y,x)\) is single-valued for every \(x\in\) dom \(g\). It is still unknown whether a Chebyshev subset of a Hilbert space \(X\) is convex or not. If \(X\) is finite-dimensional, then the answer is yes, a result that also holds in finite-dimensional smooth normed spaces. Some positive results, strongly connected with the geometric properties of the underlying Banach space, are known in infinite dimensions, too, see the survey by \textit{V.\,S.\thinspace Balaganskij} and \textit{L.\,P.\thinspace Vlasov} [Russ.\ Math.\ Surv.\ 51, No.\,6, 1127--1190 (1996; Zbl 0931.41017), translation from Usp.\ Mat.\ Nauk 51, No.\,6, 125--188 (1996); errata ibid.\ 52, No.\,1, 237 (1997)]. These results were extended by \textit{H.\,Bauschke, X.\,Wang, J.\,Ye} and \textit{X.\,Yuan} [J.~Approx.\ Theory 159, No.\,1, 3--25 (2009; Zbl 1175.49017)] to \(D_g\)-Chebyshev sets in the \(n\)-dimensional Euclidean space, supposing that the function \(g\) satisfies some adequate conditions. The aim of the present paper is to further extend some of these results to infinite-dimensional reflexive Banach spaces, giving, at the same time, generalizations of the corresponding results on the convexity of Chebyshev sets in Banach spaces as well as of some results of \textit{Y.\,I.\thinspace Alber} [Lect.\ Notes Pure Appl.\ Math.\ 178, 15--50 (1996; Zbl 0883.47083)] on generalized projection operators. A key role in this study is played by the notions of \(D_g\)-approximative …
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 49J50
Mathematics Subject Classification ID: 49J53
Mathematics Subject Classification ID: 90C25
zbMATH DE Number: 5726579
zbMATH Keywords: Bregman distance
zbMATH Keywords: Bregman projection
zbMATH Keywords: Chebyshev set
zbMATH Keywords: \(D\)-approximative compactness
zbMATH Keywords: Gâteaux differentiability
zbMATH Keywords: Fréchet differentiability
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: j.jat.2009.12.006
OpenAlex ID: W2136788324
cites work: Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications
cites work: Cebysev Sets in Hilbert Space
cites work: Gradients of Convex Functions
cites work: Approximate properties of sets in Hilbert space
cites work: Q4375224
cites work: ESSENTIAL SMOOTHNESS, ESSENTIAL STRICT CONVEXITY, AND LEGENDRE FUNCTIONS IN BANACH SPACES
cites work: Bregman Monotone Optimization Algorithms
cites work: Bregman distances and Chebyshev sets
cites work: Q4503250
cites work: Proximality and Chebyshev sets
cites work: Nonlinear maximal monotone operators in Banach space
cites work: Q5800605
cites work: Iterative averaging of entropic projections for solving stochastic convex feasibility problems
cites work: Q4336952
cites work: Totally convex functions for fixed points computation and infinite dimensional optimization
cites work: Total convexity for powers of the norm in uniformly convex Banach spaces
cites work: Q4422674
cites work: Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces
cites work: Best approximation in inner product spaces
cites work: Q3286569
cites work: Well Posed Optimization Problems and Nonconvex Chebyshev Sets in Hilbert Spaces
cites work: \(W\)-approximative compactness and continuity of the generalized projection operator in Banach spaces
cites work: Potpourri of Conjectures and Open Questions in Nonlinear Analysis and Optimization
cites work: On Johnson's example of a nonconvex Chebyshev set
cites work: A nonconvex set which has the unique nearest point property
cites work: Proximal Minimization Methods with Generalized Bregman Functions
cites work: Convexity of Chebyshev sets
cites work: Approximative compactness and continuity of metric projections
cites work: Q4669340
cites work: On the virtual convexity of the domain and range of a nonlinear maximal monotone operator
cites work: Q5552462
cites work: Almost convex and Chebyshev sets
cites work: APPROXIMATIVE PROPERTIES OF SETS IN NORMED LINEAR SPACES
cites work: On a Property of Metric Projections Onto Closed Subsets of Hilbert Spaces
cites work: On uniformly convex functions
cites work: Q3149262
DOI: 10.1016/J.JAT.2009.12.006
description: scientific article; zbMATH DE number 5726579