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/

A. Björck, Numerical methods for least squares problems. SIAM Publ. (1996). | MR | Zbl

R. Bramley and B. Winnicka, Solving linear inequalities in a least squares sense. SIAM J. Sci. Comput. 17 (1996) 275–286. | DOI | MR | Zbl

P.L. Combettes, Inconsistent signal feasibility problems: least squares solutions in a product space. IEEE Trans. Signal Process. 42 (1994) 2955–2966. | DOI

S.P. Han, Least-squares solution of linear inequalities. Technical Report TR-2141, Mathematics Research Center, University of Wisconsin-Madison (1980).

R.T. Rockafellar, Convex analysis. Princeton University Press, Second printing (1972). | MR | Zbl

J.H. Spoonamore, Least squares methods for solving systems of inequalities with application to an assignment problem. Technical Report TM FF-93/03, USACERL (1992). | MR

K. Yang, New iterative methods for linear inequalities. Technical Report 90-06, University of Michigan (1990). | MR

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