Projectionally exposed cones in \(R^ 3\)
type: Item
instance of: scholarly article
title: Projectionally exposed cones in \(R^ 3\)
zbMATH Open document ID: 0662.90062
DOI: 10.1016/0024-3795(88)90058-4
author: George D. Poole
author: Michael Laidacker
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: \textit{J. Borwein} and \textit{H. Wolkowicz} [J. Math. Anal. Appl. 83, 495-530 (1981; Zbl 0467.90076)] introduced the concept of p-exposed (projectionally exposed) cones, which can be used, if the constraint function of an optimization problem takes values in such a cone; then certain multipliers may be chosen from a smaller set. A closed and pointed cone K is called p-exposed, if every of its faces is. A face F of K is a subcone F such that \(x\in K\), \(y\in F\) and y-x\(\in K\) imply \(x\in F\). A face F of K is exposed if there is an \(f\in K^*\) such that \(F=\{x\in K:\) \(f(x)=0\}\), and p-exposed, if there is a projection P such that \(P(K)=F.\) In this paper the p-exposed cones of \(R^ 3\) are completely characterized: K is p-exposed if and only if each face of K is exposed. The authors believe that this result extends to \(R^ n\) for all n, but it seems to them, as if it is difficult to prove it.
Mathematics Subject Classification ID: 90C25
Mathematics Subject Classification ID: 52Bxx
Mathematics Subject Classification ID: 52A15
zbMATH DE Number: 4083372
zbMATH Keywords: (projectionally exposed) cones
zbMATH Keywords: p-exposed cones
reviewed by: Alfred Göpfert
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90058-4
OpenAlex ID: W2087169144
cites work: Perfect cones
cites work: Projectionally Exposed Cones
cites work: Q3943082
cites work: Regularizing the abstract convex program
cites work: Cônes autopolaires et algèbres de Jordan
cites work: Q4107157
description: scientific article; zbMATH DE number 4083372