NDLI: Minmax-distance approximation and separation problems: geometrical properties

Content Provider	Springer Nature Link
Author	Plastria, Frank Carrizosa, Emilio
Copyright Year	2010
Abstract	A center hyperplane in the d-dimensional space minimizes the maximum of its distances from a finite set of points A with respect to possibly different gauges. In this note it is shown that a center hyperplane exists which is at (equal) maximum distance from at least d + 1 points of A. Moreover the projections of the points among these which lie above the center hyperplane cannot be separated by another hyperplane from the projections of those that are below it. When all gauges involved are smooth, all center hyperplanes satisfy these properties. This geometric property allows us to improve and generalize previously existing results, which were only known for the case in which all distances are measured using a common norm. The results also extend to the constrained case where for some points it is prespecified on which side of the hyperplane (above, below or on) they must lie. In this case the number of points lying on the hyperplane plus those at maximum distance is at least d + 1. It follows that solving such global optimization problems reduces to inspecting a finite set of candidate solutions. Extensions of these results to a separation problem are outlined.
Ending Page	177
Page Count	25
Starting Page	153
File Format	PDF
ISSN	00255610
e-ISSN	14364646
Journal	Mathematical Programming
Issue Number	1-2
Volume Number	132
Language	English
Publisher	Springer-Verlag
Publisher Date	2010-06-04
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematical Methods in Physics Center hyperplane Classification and discrimination; cluster analysis Linear separation Mathematics of Computing Calculus of Variations and Optimal Control; Optimization Gauge-distance to hyperplane Numerical Analysis Theoretical, Mathematical and Computational Physics Combinatorics Minimax problems
Content Type	Text
Resource Type	Article
Subject	Mathematics Software

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