Geodesic
Least squares solutions of linear inequality systems: a pedestrian approach
Contesse, Luis1 ; Hiriart-Urruty, Jean-Baptiste2 ; Penot, Jean-Paul3
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.
RAIRO - Operations Research - Recherche Opérationnelle, Tome 51 (2017) no. 3, pp. 567-575

Voir la notice de l'article provenant de la source Numdam

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
Keywords: 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. 
@article{RO_2017__51_3_567_0,
     author = {Contesse, Luis and Hiriart-Urruty, Jean-Baptiste and Penot, Jean-Paul},
     title = {Least squares solutions of linear inequality systems: a pedestrian approach},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {567--575},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {3},
     year = {2017},
     doi = {10.1051/ro/2016042},
     mrnumber = {3880512},
     zbl = {1387.90185},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ro/2016042/}
}
TY  - JOUR
AU  - Contesse, Luis
AU  - Hiriart-Urruty, Jean-Baptiste
AU  - Penot, Jean-Paul
TI  - Least squares solutions of linear inequality systems: a pedestrian approach
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2017
SP  - 567
EP  - 575
VL  - 51
IS  - 3
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/ro/2016042/
DO  - 10.1051/ro/2016042
LA  - en
ID  - RO_2017__51_3_567_0
ER  - 
%0 Journal Article
%A Contesse, Luis
%A Hiriart-Urruty, Jean-Baptiste
%A Penot, Jean-Paul
%T Least squares solutions of linear inequality systems: a pedestrian approach
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2017
%P 567-575
%V 51
%N 3
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/ro/2016042/
%R 10.1051/ro/2016042
%G en
%F RO_2017__51_3_567_0
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

Cité par Sources :