Critical angles in polyhedral convex cones: Numerical and statistical considerations
type: Item
instance of: scholarly article
title: Critical angles in polyhedral convex cones: Numerical and statistical considerations
zbMATH Open document ID: 1188.52004
author: Daniel Gourion
author: Alberto Seeger
published in: Mathematical Programming. Series A. Series B
publication date: 2010-02-25T00:00:00Z
review text: The maximal angle of a nontrivial closed convex cone \(K\) in Euclidean space \({\mathbb R}^n\) is defined as the nonnegative number \(\theta_{\max}(K) = \max_{u,v\in K\cap {\mathbb S}_n}\arccos\langle u, v\rangle\), where \(\langle u, v\rangle\) is the usual inner product, while \({\mathbb S}_n\) is the unit sphere in \({\mathbb R}^n\). \(K\) is nontrivial, if \(K\) is different from \(\{0\}\) and from the whole space \({\mathbb R}^n\). \((u, v)\) is a critical pair of \(K\) means that \(v-\langle u, v\rangle\,u\in K^+\) and \(u-\langle u, v\rangle\,v\in K^+\), where the symbol \(K^+\) stands for the positive dual cone of \(K\). A critical pair \((u, v)\) and the corresponding critical angle \(\theta(u, v)= \arccos\langle u, v\rangle\) are said to be proper if \(u\) and \(v\) are not collinear. The set of all proper critical angles of \(K\) is called the angular spectrum of \(K\). The number \(\theta_{\max}(K)\) is the largest critical angle of \(K\). This critical angle is proper if and only if \(K\) is pointed and not a ray. Special attention is paid to the class of polyhedral convex cones because the angular spectrum of a convex cone of the form \(K= cone\{g^1,\dots, g^p\}\) is a finite set (see [1], Theorem 8.5) [\textit{A. Iusem} and \textit{A. Seeger}, Math. Program. 120, No.~1 (B), 3--25 (2009; Zbl 1163.52003)] and, what is even important, it can be computed explicitly by solving a series of generalized eigenvalue problems involving the vectors \(g^1, \dots, g^p\). The paper is organized as follows. A necessary technical background is provided by Section 2 titled ``Preliminary results''. Throughout Section 3 titled ``Numerical experimentation'', one assumes that \(K\) is a polyhedral convex cone in \({\mathbb R}^n\) described by a generator set \(\{g^1,\dots, g^p\}\). The dimension \(n\) of the underlying Euclidean space is assumed to be at least 3. Angular spectra are computed by solving appropriate generalized eigenvalue problems. In particular, in subsect…
Mathematics Subject Classification ID: 52A22
Mathematics Subject Classification ID: 90C26
zbMATH DE Number: 5675096
zbMATH Keywords: generalized eigenvalue problem
zbMATH Keywords: generators of the cone
zbMATH Keywords: local maximal angles
zbMATH Keywords: random polyhedral convex cone
reviewed by: Viktor Ohanyan
MaRDI profile type: Publication
full work available at URL: s10107-009-0317-2
OpenAlex ID: W1965434604
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DOI: 10.1007/S10107-009-0317-2
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description: scientific article; zbMATH DE number 5675096