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Gauge Distances and Median Hyperplanes

Author

Listed:
  • F. Plastria

    (Vrije Universiteit)

  • E. Carrizosa

    (Universidad de Sevilla)

Abstract

A median hyperplane in d-dimensional space minimizes the weighted sum of the distances from a finite set of points to it. When the distances from these points are measured by possibly different gauges, we prove the existence of a median hyperplane passing through at least one of the points. When all the gauges are equal, some median hyperplane will pass through at least d-1 points, this number being increased to d when the gauge is symmetric, i.e. the gauge is a norm.Whereas some of these results have been obtained previously by different methods, we show that they all derive from a simple formula for the distance of a point to a hyperplane as measured by an arbitrary gauge.

Suggested Citation

  • F. Plastria & E. Carrizosa, 2001. "Gauge Distances and Median Hyperplanes," Journal of Optimization Theory and Applications, Springer, vol. 110(1), pages 173-182, July.
    • Handle: RePEc:spr:joptap:v:110:y:2001:i:1:d:10.1023_a:1017551731021
    DOI: 10.1023/A:1017551731021
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    References listed on IDEAS

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    1. Durier, Roland & Michelot, Christian, 1985. "Geometrical properties of the Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 20(3), pages 332-343, June.
    2. Zvi Drezner & George O. Wesolowsky, 1989. "The Asymmetric Distance Location Problem," Transportation Science, INFORMS, vol. 23(3), pages 201-207, August.
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    Cited by:

    1. Marc Ciligot-Travain & Sado Traoré, 2014. "On a robustness property in single-facility location in continuous space," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 321-330, April.
    2. Emilio Carrizosa & Frank Plastria, 2008. "Optimal Expected-Distance Separating Halfspace," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 662-677, August.
    3. Jack Brimberg & Henrik Juel & Mark-Christoph Körner & Anita Schöbel, 2014. "Locating an axis-parallel rectangle on a Manhattan plane," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 185-207, April.
    4. Carrizosa, Emilio & Goerigk, Marc & Schöbel, Anita, 2017. "A biobjective approach to recoverable robustness based on location planning," European Journal of Operational Research, Elsevier, vol. 261(2), pages 421-435.
    5. Frank Plastria & Steven De Bruyne & Emilio Carrizosa, 2010. "Alternating local search based VNS for linear classification," Annals of Operations Research, Springer, vol. 174(1), pages 121-134, February.
    6. Diaz-Banez, J. M. & Mesa, J. A. & Schobel, A., 2004. "Continuous location of dimensional structures," European Journal of Operational Research, Elsevier, vol. 152(1), pages 22-44, January.
    7. Rafael Blanquero & Emilio Carrizosa & Pierre Hansen, 2009. "Locating Objects in the Plane Using Global Optimization Techniques," Mathematics of Operations Research, INFORMS, vol. 34(4), pages 837-858, November.
    8. Baldomero-Naranjo, Marta & Martínez-Merino, Luisa I. & Rodríguez-Chía, Antonio M., 2020. "Tightening big Ms in integer programming formulations for support vector machines with ramp loss," European Journal of Operational Research, Elsevier, vol. 286(1), pages 84-100.
    9. Sönke Behrends & Anita Schöbel, 2020. "Generating Valid Linear Inequalities for Nonlinear Programs via Sums of Squares," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 911-935, September.
    10. Kristian Sabo & Rudolf Scitovski & Ivan Vazler, 2011. "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 Dat," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 293-314, May.
    11. Jack Brimberg & Robert Schieweck & Anita Schöbel, 2015. "Locating a median line with partial coverage distance," Journal of Global Optimization, Springer, vol. 62(2), pages 371-389, June.

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