On stable least squares solution to the system of linear inequalities

• Autorzy: Evald Übi
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 65K05: Mathematical programming methods
• 90C05: Linear programming

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2007
• Tom: 5
• Numer: 2
• Strony: 373-385

Daty

wydano

2007-06-01

online

2007-01-26

Twórcy

autor

Evald Übi

• Tallinn University of Technology

Bibliografia

• [1] C. Lawson and R. Hanson: Solving Least Squares Problems, Prentice-Hall, New-Jersey, 1974.
• [2] E. Übi: “Exact and Stable Least Squares Solution to the Linear Programming Problem”, Centr. Eur. J. Math., Vol. 3(2), (2005), pp. 228–241. http://dx.doi.org/10.2478/BF02479198
• [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.
• [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.
• [11] A. Björck: “Generalized and Sparse Least Squares Problems”, NATO ASI Series C, Vol. 434, (1994), pp. 37–80.
• [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
• [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.

Identyfikatory

DOI

10.2478/s11533-007-0003-7