Abstract
Abstract. The anchored hyperplane location problem is to locate a hyperplane passing through some given points P \subseteq R n and minimizing either the sum of weighted distances (median problem ), or the maximum weighted distance (center problem ) to some other points Q \subseteq R n . This problem of computational geometry is analyzed by using nonlinear programming techniques. If the distances are measured by a norm, it will be shown that in the median case there exists an optimal hyperplane that passes through at least n - k affinely independent points of Q , if k is the maximum number of affinely independent points of P . In the center case, there exists an optimal hyperplane which is at maximum distance to at least n- k +1 affinely independent points of Q . Furthermore, if the norm is a smooth norm, all optimal hyperplanes satisfy these criteria. These results generalize known results about unrestricted hyperplane location problems.
Highlights
Approximating a set of given points Q in Rn by a linear function is known as the linear fit problem or the hyperplane location problem
If the distances are measured by a norm, it will be shown that in the median case there exists an optimal hyperplane that passes through at least n − k affinely independent points of Q, if k is the maximum number of affinely independent points of P
There always exists an optimal hyperplane for the median problem that passes through n affinely independent points of Q, and, in the center case, there always exists an optimal hyperplane which is at maximum distance from n + 1 affinely independent points of Q
Summary
Approximating a set of given points Q in Rn by a linear function is known as the linear fit problem or the hyperplane location problem. The goal is to find a hyperplane (represented by a linear function) minimizing the sum of weighted distances to the points in Q, or minimizing the maximum weighted distance to the points in Q, respectively. In this paper a restricted version of the hyperplane location problem—the so-called anchored hyperplane location problem—is analyzed, namely, the hyperplane is forced to pass through some given points p ∈ P. Hyperplane location problems appear in several mathematical disciplines, where they have mainly been studied with the Euclidean and the rectangular distance. In robust statistics variants of the hyperplane location problem are known as absolute errors regression, median problems, L1 regression, L∞ regression, and orthogonal/vertical
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