Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

Kristian Sabo (), Rudolf Scitovski () and Ivan Vazler ()
Additional contact information
Kristian Sabo: University of Osijek
Rudolf Scitovski: University of Osijek
Ivan Vazler: University of Osijek

Journal of Optimization Theory and Applications, 2011, vol. 149, issue 2, No 4, 293-314

Abstract: Abstract We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈ℝm×n, m≥n, $\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}$ . This problem is equivalent to the problem of determining a best LAD-hyperplane x↦a T x, x∈ℝ n on the basis of given data $(\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m$ , 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 a ∗∈ℝ n is the point of a global minimum of the functional F if and only if 0∈∂F(a ∗). Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)↦αx+βy, passing through the origin of the coordinate system, on the basis of the data (x i ,y i ,z i ),i=1,…,m.

Keywords: LAD; Least absolute deviations; Overdetermined system of linear equations; l1-norm approximation; Weighted median problem; Outliers; LAD-hyperplane (search for similar items in EconPapers)
Date: 2011
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10957-010-9791-1 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:149:y:2011:i:2:d:10.1007_s10957-010-9791-1

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-010-9791-1

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().