Bregman distances and Chebyshev sets

JOURNAL OF APPROXIMATION THEORY

卷:159

Article

Bauschke, Heinz H.¹  Wang, Xianfu¹  Ye, Jane²  Yuan, Xiaoming³ 

[1] Univ British Columbia Okanagan, Irving K Barber Sch, Kelowna, BC V1V 1V7, Canada

[2] Univ Victoria, Dept Math & Stat, Victoria, BC V8W 3P4, Canada

[3] Hong Kong Baptist Univ, Dept Math, Hong Kong, Hong Kong, Peoples R China

关键词: Bregman distance;    Bregman projection;    Chebyshev set with respect to a Bregman distance;    Legendre function;    Maximal monotone operator;    Nearest point;    Subdifferential operators;   

DOI  :  10.1016/j.jat.2008.08.014

来源: Elsevier

【 摘 要 】

A closed set of a Euclidean space is said to be Chebyshev if every point in the space has one and only one closest point in the set. Although the situation is not settled in infinite-dimensional Hilbert spaces, in 1932 Built showed that in Euclidean spaces a closed set is Chebyshev if and only if the set is convex. In this paper, from the more general perspective of Bregman distances, we show that if every point in the space has a unique nearest point in a closed set, then the set is convex. We provide two approaches: one is by nonsmooth analysis; the other by maximal monotone operator theory. Subdifferentiability properties of Bregman nearest distance functions are also given. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free