# Exact and stable least squares solution to the linear programming problem

Open Mathematics (2005)

• Volume: 3, Issue: 2, page 228-241
• ISSN: 2391-5455

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## Abstract

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A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A. Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.

## How to cite

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Evald Übi. "Exact and stable least squares solution to the linear programming problem." Open Mathematics 3.2 (2005): 228-241. <http://eudml.org/doc/268717>.

@article{EvaldÜbi2005,
abstract = {A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A. Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.},
author = {Evald Übi},
journal = {Open Mathematics},
keywords = {90C05; 65K05},
language = {eng},
number = {2},
pages = {228-241},
title = {Exact and stable least squares solution to the linear programming problem},
url = {http://eudml.org/doc/268717},
volume = {3},
year = {2005},
}

TY - JOUR
AU - Evald Übi
TI - Exact and stable least squares solution to the linear programming problem
JO - Open Mathematics
PY - 2005
VL - 3
IS - 2
SP - 228
EP - 241
AB - A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A. Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered.
LA - eng
KW - 90C05; 65K05
UR - http://eudml.org/doc/268717
ER -

## References

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1. [1] E. Übi: “An Approximate Solution to Linear and Quadratic Programming Problems by the Method of least squares”, Proc. Estonian Acad. Sci. Phys. Math., Vol. 47, (1998), pp. 19–28. Zbl0969.90065
2. [2] E. Übi: “On Computing a Stable Least Squares Solution to the Linear Programming Problem”, Proc. Estonian Acad. Sci. Phys. Math., Vol 47, (1998), pp. 251–259. Zbl1058.90513
3. [3] E. Übi: “Finding Non-negative Solution of Overdetermined or Underdetermined System of Linear Equations by Method of Least Squares”, Trans. Tallinn Tech. Univ., Vol. 738, (1994), pp. 61–68.
4. [4] R. Cline and R. Plemmons: l 2-solutions to Underdetermined Linear Systems SIAM Review, Vol. 10, (1976), pp. 92–105. http://dx.doi.org/10.1137/1018004
5. [5] A. Cline: “An Elimination Method for the Solution of Linear Least Squares Problems”, SIAM J. Numer. Anal., Vol. 10, (1973), pp. 283–289. http://dx.doi.org/10.1137/0710027 Zbl0253.65023
6. [6] C. Lawson and R. Hanson: Solving Least Squares Problems, Prentice-Hall, New-Jersey, 1974. Zbl0860.65028
7. [7] B. Poljak: Vvedenie v optimizatsiyu, Nauka, Moscow, 1983.
8. [8] T. Hu: Integer programming and Network flows, Addison-Wesley Publishing Company, Massachusetts, 1970.

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