Duality and Lipschitzian selections in best approximation from nonconvex cones
type: Item
instance of: scholarly article
title: Duality and Lipschitzian selections in best approximation from nonconvex cones
zbMATH Open document ID: 0746.41029
DOI: 10.1016/0021-9045(91)90066-J
author: V. A. Ubhaya
published in: Journal of Approximation Theory
publication date: 1992-06-25T00:00:00Z
review text: The author's abstract states: ``Duality relationships in finding a best approximation from a nonconvex cone in a normed linear space in general, and in the space of bounded functions in particular, are investigated. The cone and the dual problems are defined in terms of positively homogeneous super-additive functionals on the space. Conditions are developed on the cone so that the duality gap between a pair of primal and dual problems does not exist. In addition, Lipschitz continuous selections of the metric projection are identified. The result are specialized to a convex cone. Applications are indicated to approximation problems.'' The paper is concerned with spaces \(B\) of real functions \(f\) on a set \(S\) which are such that \(wf\) is bounded, where \(w\) is a fixed positive weight function, and best approximation in \(B\) from a cone \(K\). A space \(B\) is normed by \(\| f\|=\sup| f| w\). The general results obtained are applied to the following examples: (4.1) \(S\) a convex subset of a vector space and \(K\) the set of all quasi-convex functions in \(B\); (4.2) \(S\) a convex subset of a vector space and \(K\) the set of all convex functions in \(B\); (4.3) \(S\) a proper open interval and \(K\) the set of all sub-additive functions in \(B\); and (5.1) certain generalised problems of monotone approximation with constraints. The paper is closely related to earlier work of the author [e.g., ibid. 55, No. 3, 326-336 (1988; Zbl 0699.41018) and ibid. 56, No. 2, 217-224 (1989; Zbl 0673.41027)].
Mathematics Subject Classification ID: 41A50
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 8513
zbMATH Keywords: Lipschitz continuous selections
zbMATH Keywords: metric projection
zbMATH Keywords: monotone approximation with constraints
reviewed by: Aldric L. Brown
MaRDI profile type: Publication
cites work: Some function classes related to the class of convex functions
cites work: Q3850015
cites work: Applications of the Hahn-Banach Theorem in Approximation Theory
cites work: A survey of metric selections
cites work: Q3778432
cites work: Characterizations of continuous and Lipschitz continuous metric selections in normed linear spaces
cites work: Q3254057
cites work: Q3288313
cites work: Q3244654
cites work: Smoothness of approximation
cites work: Analytic Inequalities
cites work: Q3667345
cites work: Another Proof that Convex Functions are Locally Lipschitz
cites work: Q5638112
cites work: Subadditive functions
cites work: Q5595759
cites work: Duality in approximation and conjugate cones in normed linear spaces
cites work: Generalized isotone optimization with applications to starshaped functions
cites work: Quasi-convex optimization
cites work: Uniform approximation by quasi-convex and convex functions
cites work: Lipschitzian selections in approximation from nonconvex sets of bounded functions
cites work: Lipschitzian selections in best approximation by continuous functions
cites work: Best quasi-convex uniform approximation
cites work: A duality approach to best uniform convex approximation
full work available at URL: 0021-9045(91)90066-j
OpenAlex ID: W2070686460
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description: scientific article; zbMATH DE number 8513