DOI: 10.2478/s11533-007-0003-7 Research article CEJM 5(2) 2007 373–385 On stable least squares solution to the system of linear inequalities Evald ¨ Ubi ∗ Department of Economics, Tallinn University of Technology, 11712 Tallinn, Estonia Received 8 May 2006; accepted 30 December 2006 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. c  Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: System of linear inequalities, method of least squares, Householder transformation, successive projection MSC (2000): 90C05, 65K05 1 Introduction During the last 50 years the simplex method based on Gaussian elimination is used to solve most of the linear programming and related problems. But for some problems the simplex method is having poor results. 30 years ago more through investigation of such problems was started. The simplex method and the polynomial-time interior point method give poor results in solving unstable linear and quadratic programming problems. For such problems the least squares method based on orthogonal transformations is more recommendable, because there is no change of vectors’ norm. The purpose of this paper is solving a system of inequalities using highly developed least squares technique. This method is used not only in mathematics but also in statis- * E-mail: evaldy@tv.ttu.ee