Abstract
We consider the following problem. Given a finite set of pointsy j in\(\mathbb{R}^n \) we want to determine a hyperplane H such that the maximum Euclidean distance betweenH and the pointsy j is minimized. This problem(CHOP) is a non-convex optimization problem with a special structure. Forexample, all local minima can be shown to be strongly unique. We present agenericity analysis of the problem. Two different global optimizationapproaches are considered for solving (CHOP). The first is a Lipschitzoptimization method; the other a cutting plane method for concaveoptimization. The local structure of the problem is elucidated by analysingthe relation between (CHOP) and certain associated linear optimizationproblems. We report on numerical experiments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
U. Faigle, W. Kern and M. Streng, Note on the computational complexity of j-radii of polytopes in ℝn, Mathematical Programming73, 1–5, 1996.
P. Gritzmann and V. Klee, Inner and outer j-radii of convex bodies in finite dimensional normed spaces, Discrete Computational Geometry7, 255–280, 1992.
R. Hettich and P. Zencke, Numerische Methoden der Approximation und Semi-infiniten Optimierung, B.G. Teubner, 1982.
R. Horst and H. Tuy, Global Optimization, Springer Verlag, 1990.
C.G. Gibson, K. Wirthmüller, A.A. Du Plessis and E.J.N. Looijenga, Topological Stability of Smooth Mappings, Lecture Notes in Mathematics, Vol. 522, Springer Verlag, Berlin, 1976.
F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays: Courant Anniversary Volume, Interscience publishers, New York, 187–204, 1948.
H.W. Jung, Ueber die kleinste kugel, die eine räumliche Figur einschliesst, Journal für die reine und angewandte Mathematik123, 241–257, 1901.
P. Jonker, M. Streng and F. Twilt, Chebyshev Line Approximation: a Sensitivity Analysis, in Parametric Optimization and Related Topics III, J. Guddat, H. Th. Jongen, B. Kummer and F. Nočička (eds.) (Approximation and Optimization), Peter Lang Verlag, Frankfurt am Main, 303–322, 1993.
D. Klatte, On quantitative stability for non-linear minima, Control and Cybernetics23(1/2), 183–200, 1994.
H. Späth, Orthogonal least squares fitting with linear manifolds, Numerische Mathematik86, 441–445, 1986.
G. Still and M. Streng, Optimality Conditions in Smooth Nonlinear Optimization, Journal of Optimization Theory and Applications90(3), 483–516, 1996.
M. Streng, Chebyshev Approximation of Finite Sets by Curves and Linear Manifolds, dissertation, University of Twente, 1993.
M. Streng and W. Wetterling, Chebyshev Approximation of a Point Set by a Straight Line, Constructive Approximation10, 187–196, 1994.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Still, G., Streng, M. The Chebyshev Hyperplane Optimization Problem. Journal of Global Optimization 11, 361–376 (1997). https://doi.org/10.1023/A:1008220431204
Issue Date:
DOI: https://doi.org/10.1023/A:1008220431204