Least squares solutions of linear inequality systems: a pedestrian approach
RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 3, pp. 567-575.

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.

Reçu le :
Accepté le :
DOI : 10.1051/ro/2016042
Classification : 90C25, 93E24, 52A40, 65K10
Mots clés : Linear inequalities, least squares solutions, convex polyhedron, quadratic function, alternative theorem
Contesse, Luis 1 ; Hiriart-Urruty, Jean-Baptiste 2 ; Penot, Jean-Paul 3

1 Facultad de Ingenieria, Pontificia Universidad Catolica de Chile, Santiago, Chile.
2 Institut de Mathématiques, Université Paul Sabatier, Toulouse, France
3 Laboratoire J.L. Lions, Université P. et M. Curie, Paris, France.
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     title = {Least squares solutions of linear inequality systems: a pedestrian approach},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {567--575},
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Contesse, Luis; Hiriart-Urruty, Jean-Baptiste; Penot, Jean-Paul. Least squares solutions of linear inequality systems: a pedestrian approach. RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 3, pp. 567-575. doi : 10.1051/ro/2016042. http://www.numdam.org/articles/10.1051/ro/2016042/

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