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Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data (CROSBI ID 167379)

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Sabo, Kristian ; Scitovski, Rudolf ; Vazler, Ivan Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data // Journal of optimization theory and applications, 149 (2011), 2; 293-314. doi: 10.1007/s10957-010-9791-1

Podaci o odgovornosti

Sabo, Kristian ; Scitovski, Rudolf ; Vazler, Ivan

engleski

Searching for a best LAD-solution of an overdetermined system of linear equations motivated by searching for a best LAD-hyperplane on the basis of given data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations $\mathbf{; ; ; ; Xa}; ; ; ; =\mathbf{; ; ; ; z}; ; ; ; $, $\mathbf{; ; ; ; X}; ; ; ; \in\R^{; ; ; ; m\times n}; ; ; ; $, $m\geq n$, $\mathbf{; ; ; ; a}; ; ; ; \in\R^n, \mathbf{; ; ; ; z}; ; ; ; \in\R^m$. This problem is equivalent to the problem of determining a best LAD-hyperplane $\mathbf{; ; ; ; x}; ; ; ; \mapsto \mathbf{; ; ; ; a}; ; ; ; ^T\mathbf{; ; ; ; x}; ; ; ; $, $\mathbf{; ; ; ; x}; ; ; ; \in\R^n$ on the basis of given data $(\mathbf{; ; ; ; x}; ; ; ; _i, z_i), \, \mathbf{; ; ; ; x}; ; ; ; _i=(x_1^{; ; ; ; (i)}; ; ; ; , \ldots, x_n^{; ; ; ; (i)}; ; ; ; )^T\in\R^n, \, z_i\in\R, \, i=1, \ldots, m$, whereby the minimizing functional is of the form \[ F(\mathbf{; ; ; ; a}; ; ; ; )=\|\mathbf{; ; ; ; z}; ; ; ; -\mathbf{; ; ; ; Xa}; ; ; ; \|_1=\sum_{; ; ; ; i=1}; ; ; ; ^m|z_i-\mathbf{; ; ; ; a}; ; ; ; ^T\mathbf{; ; ; ; x}; ; ; ; _i|. \] An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional $F$ is convex, and therefore the point $\mathbf{; ; ; ; a}; ; ; ; ^*\in\R^n$ is the point of a global minimum of the functional $F$ if and only if $\mathbf{; ; ; ; 0}; ; ; ; \in\partial F(\mathbf{; ; ; ; a}; ; ; ; ^*)$. Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane $(x, y)\mapsto \alpha x+\beta y$, passing through the origin of the coordinate system, on the basis of the data $(x_i, y_i, z_i), \, i=1, \ldots, m$.

LAD; least absolute deviations; overdetermined system of linear equations; $l_1$-norm approximation; weighted median problem; outliers; LAD-hyperplane

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Podaci o izdanju

149 (2)

2011.

293-314

objavljeno

0022-3239

10.1007/s10957-010-9791-1

Povezanost rada

Matematika

Poveznice
Indeksiranost