arXiv subjects

Xianfu Wang

Publications and source records attributed to Xianfu Wang.

93 records · Page 6Linked to original sources

arXiv2008

Bregman distances and Klee sets

In 1960, Klee showed that a subset of a Euclidean space must be a singleton provided that each point in the space has a unique farthest point in the set. This classical result has received much attention; in fact, the Hilbert space version is a famous open problem. In this paper, we consider Klee sets from a new perspective. Rather than measuring distance induced by a norm, we focus on the case when distance is meant in the sense of Bregman, i.e., induced by a convex function. When the convex function has sufficiently nice properties, then - analogously to the Euclidean distance case - every Klee set must be a singleton. We provide two proofs of this result, based on Monotone Operator Theory and on Nonsmooth Analysis. The latter approach leads to results that complement work by Hiriart-Urruty on the Euclidean case.

math.FA↗

arXiv2008

Autoconjugate representers for linear monotone operators

Monotone operators are of central importance in modern optimization and nonlinear analysis. Their study has been revolutionized lately, due to the systematic use of the Fitzpatrick function. Pioneered by Penot and Svaiter, a topic of recent interest has been the representation of maximal monotone operators by so-called autoconjugate functions. Two explicit constructions were proposed, the first by Penot and Zalinescu in 2005, and another by Bauschke and Wang in 2007. The former requires a mild constraint qualification while the latter is based on the proximal average. We show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces. The continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate. Furthermore, we also construct an infinite family of autoconjugate representers for the identity operator on the real line.

math.FA↗

arXiv2007

Bregman distances and Chebyshev sets

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 Bunt 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.

math.FA↗