# On stable least squares solution to the system of linear inequalities

Open Mathematics (2007)

- Volume: 5, Issue: 2, page 373-385
- ISSN: 2391-5455

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topEvald Übi. "On stable least squares solution to the system of linear inequalities." Open Mathematics 5.2 (2007): 373-385. <http://eudml.org/doc/269489>.

@article{EvaldÜbi2007,

abstract = {The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author’s previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable and degenerate problems primarily.},

author = {Evald Übi},

journal = {Open Mathematics},

keywords = {System of linear inequalities; method of least squares; Householder transformation; successive projection},

language = {eng},

number = {2},

pages = {373-385},

title = {On stable least squares solution to the system of linear inequalities},

url = {http://eudml.org/doc/269489},

volume = {5},

year = {2007},

}

TY - JOUR

AU - Evald Übi

TI - On stable least squares solution to the system of linear inequalities

JO - Open Mathematics

PY - 2007

VL - 5

IS - 2

SP - 373

EP - 385

AB - The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author’s previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable and degenerate problems primarily.

LA - eng

KW - System of linear inequalities; method of least squares; Householder transformation; successive projection

UR - http://eudml.org/doc/269489

ER -

## References

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- [3] Fan Ky: “On Systems of Linear Inequalities”, In: H. Kuhn and W. Tucker (Eds.): Linear Inequalities and Related Systems, Priceton, 1956.
- [4] E. Übi: “Finding Non-negative Solution of Overdetermined or Underdetermined System of Linear Equations by Method of Least Squares”, Transactions of Tallinn TU, Vol. 738, (1994), pp. 61–68.
- [5] C. Papadimitriou and K. Steiglitz: Combinatorial Optimization:Algorithms and Complexity, Prentice-Hall, New-Jersey, 1982. Zbl0503.90060
- [6] L. Khachiyan: “A Polynomial Algorihm in linear programming”, Soviet Mathematics Doklady, Vol. 20, (1979), pp. 191–194.
- [7] L. Khachiyan: “Fourier-Motzkin Elimination Method”, Encyklopedia of Optimization, Vol. 2, (2001), pp. 155–159.
- [8] G. Danzig: Linear Programming and Extensions, Princeton University Press, 1963.
- [9] S. Chernikov: Lineare Ungleichungen, Deutscher Verlag der Wissenschaften, Berlin, 1971.
- [10] D. Gale: The Theory of Linear Economic Models, McGgraw-HILL Book Company, 1960. Zbl0114.12203
- [11] A. Björck: “Generalized and Sparse Least Squares Problems”, NATO ASI Series C, Vol. 434, (1994), pp. 37–80. Zbl0828.90100
- [12] M. Hath: “Some Extensions of an algorithm for Sparse Linear Least Squares Problems”, SIAM J.Sci. Statist. Comput., Vol. 3, (1982), pp. 223–237. http://dx.doi.org/10.1137/0903014 Zbl0483.65027
- [13] L. Bregman: “The Method of Successive Projecton for Finding The Common Point of Convex Set”, Soviet. Math. Dokl., Vol. 6, (1969), pp. 688–692. Zbl0142.16804

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