Abstract
The least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.
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References
Barnes E., Chen V., Gopalakrishnan B., Johnson E.L., A least-squares primal-dual algorithm for solving linear programming problems, Oper. Res. Lett., 2002, 30(5), 289–294
Bixby R.E., Implementing the simplex method: The initial basis, ORSA J. Comput., 1992, 4(3), 267–284
Dantzig G.B., Orden A., Wolfe P., The generalized simplex method for minimizing a linear form under linear inequality restraints, Pacific J. Math., 1955, 5, 183–195
Gale D., How to solve linear inequalities, Amer. Math. Monthly, 1969, 76(6), 589–599
Gill P.E., Murray W., Wright M.H., Practical Optimization, Academic Press, London, 1981
Lawson C.L., Hanson R.J., Solving Least Squares Problems, Classics in Applied Mathematics, 15, SIAM, Philadelphia, 1995
Leichner S.A., Dantzig G. B., Davis J.W., A strictly improving linear programming Phase I algorithm, Ann. Oper. Res., 1993, 46–47(2), 409–430
Netlib LP Test Problem Set, available at www.numerical.rl.ac.uk/cute/netlib.html
Kong S., Linear Programming Algorithms Using Least-Squares Method, Ph.D. thesis, Georgia Institute of Technology, Atlanta, USA, 2007
Übi E., Least squares method in mathematical programming, Proc. Estonian Acad. Sci. Phys. Math., 1989, 38(4), 423–432, (in Russian)
Übi E., An approximate solution to linear and quadratic programming problems by the method of least squares, Proc. Estonian Acad. Sci. Phys. Math., 1998, 47(4), 19–28
Übi E., On computing a stable least squares solution to the linear programming problem, Proc. Estonian Acad. Sci. Phys. Math., 1998, 47(4), 251–259
Übi E., Application of orthogonal transformations in the revised simplex method, Proc. Estonian Acad. Sci. Phys. Math., 2001, 50(1), 34–41
Übi E., On stable least squares solution to the system of linear inequalities, Cent. Eur. J. Math., 2007, 5(2), 373–385
Übi E., A numerically stable least squares solution to the quadratic programming problem, Cent. Eur. J. Math., 2008, 6(1), 171–178
Zoutendijk G., Mathematical Programming Methods, North Holland, Amsterdam-New York-Oxford, 1976
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Übi, E. Mathematical programming via the least-squares method. centr.eur.j.math. 8, 795–806 (2010). https://doi.org/10.2478/s11533-010-0049-9
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DOI: https://doi.org/10.2478/s11533-010-0049-9