1. Least squares solutions of linear inequality systems: a pedestrian approach
- Author
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Luis Contesse, Jean-Paul Penot, Jean-Baptiste Hiriart-Urruty, Departamento de Matemáticas [Santiago de Chile], Facultad de Matemáticas [Santiago de Chile], Pontificia Universidad Católica de Chile (UC)-Pontificia Universidad Católica de Chile (UC), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Mathematical optimization ,Context (language use) ,010103 numerical & computational mathematics ,Management Science and Operations Research ,01 natural sciences ,Least squares ,Theoretical Computer Science ,Iteratively reweighted least squares ,90C25, 93E24, 52A40, 65K10 ,Linear inequalities ,least squares solutions ,Least squares support vector machine ,Applied mathematics ,[MATH]Mathematics [math] ,0101 mathematics ,Total least squares ,quadratic function ,Mathematics ,alternative theorem ,convex polyhedron ,Computer Science Applications ,010101 applied mathematics ,Nonlinear system ,Linear inequality ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Linear least squares - Abstract
International audience; With the help of elementary results and techniques from Real Analysis and Optimization at the undergraduate level, we study least squares solutions of linear inequality systems. We prove existence of solutions in various ways, provide a characterization of solutions in terms of nonlinear systems, and illustrate the applicability of results as a mathematical tool for checking the consistency of a system of linear inequalities and for proving "theorems of alternative" like the one by Gordan. Since a linear equality is the conjunction of two linear inequalities, the proposed results cover and extend what is known in the classical context of least squares solutions of linear equality systems.; "De tous les principes qu'on peut proposer pour cet objet, je pense qu'il n'en est pas de plus général, de plus exact, ni d'une application plus facile que celui qui consistè a rendre minimum la somme des carrés des erreurs" "Of all the principles that can be proposed, I think there is none more general, more exact, and more easy of application, than that which consists of minimizing the sum of the squares of the errors" A.-M.Legendre, Nouvelles méthodes pour la détermination des orbites des comètes, Paris (1805).
- Published
- 2017