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  1. 3 Open Mathematics

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  1. 3 Übi E.

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  1. 1 2010

  2. 1 2007

  3. 1 2005

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Mathematical programming via the least-squares method

100%
Übi E.
Open Mathematics
|
2010
|
tom 8
|
nr 4
795-806
EN
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.
2
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On stable least squares solution to the system of linear inequalities

100%
Übi E.
Open Mathematics
|
2007
|
tom 5
|
nr 2
373-385
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.
3
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Exact and stable least squares solution to the linear programming problem

100%
Übi E.
Open Mathematics
|
2005
|
tom 3
|
nr 2
228-241
EN
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.
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