Critical angles between two convex cones I. General theory

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• 作者：Alberto Seeger ; David Sossa
• 关键词：Maximal angle ; Critical angle ; Principal angle ; Convex cone ; Canonical analysis ; Nonconvex optimization ; Optimality conditions
• 刊名：TOP
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：24
• 期：1
• 页码：44-65
• 全文大小：519 KB
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• 作者单位：Alberto Seeger (1)
  David Sossa (2)
  
  1. Département de Mathématiques, Université d’Avignon, 33 rue Louis Pasteur, 84000, Avignon, France
  2. Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático (CNRS UMI 2807), FCFM, Universidad de Chile, Blanco Encalada, 2120, Santiago, Chile
  
• 刊物主题：Operations Research/Decision Theory; Optimization; Statistics for Business/Economics/Mathematical Finance/Insurance; Industrial and Production Engineering; Game Theory/Mathematical Methods;
• 出版者：Springer Berlin Heidelberg
• ISSN：1863-8279

文摘

The concept of critical (or principal) angle between two linear subspaces has applications in statistics, numerical linear algebra, and other areas. Such concept has been abundantly studied in the literature, both from a theoretical and computational point of view. Part I of this work is an attempt to build a general theory of critical angles for a pair of closed convex cones. The need of such theory is motivated, among other reasons, by some specific problems arising in regression analysis of cone-constrained data, see Tenenhaus (Psychometrika 53:503–524, 1988). Angle maximization and/or angle minimization problems involving a pair of convex cones are at the core of our discussion. Such optimization problems are nonconvex in general and their numerical resolution offer a number of challenges. Part II of this work focusses on the practical computation of the maximal and/or minimal angle between specially structured cones.