Continuous location of dimensional structures

Díaz-Báñez, José Miguel; Mesa, Juan A.; Schöbel, Anita

doi:10.1016/s0377-2217(02)00647-1

Cited by 54 publications

(27 citation statements)

References 88 publications

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“…This proves existence of an optimal solution to (11), and hence, by equivalence, of an optimal solution (u * , β * ) to Problem (4). Hence, H (u * , β * ) is a constrained center hyperplane.…”

Section: Theorem 2 There Always Exists a Constrained Center Hyperplanementioning

confidence: 52%

“…[11,14,15,19,23,24,36,37,39], first for Euclidean distances, later for a general norm distance. Some geometrical properties have been obtained for these cases and will be generalized here, both in form and in application range.…”

Section: Gauge Distances and Center Hyperplanesmentioning

confidence: 99%

“…sphere, hyperplane or polygon in the plane), leading to location models for extensive facilities. See [2,11,36] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…[3,4,11,12,20,25,27,37,38,41], but are also considered in other fields. In Approximation Theory and Statistics orthogonal regression calls for finding a hyperplane minimizing some function of its distances to given data-points [14,15,18,19,24,29,32,39].…”

Section: Introductionmentioning

confidence: 99%

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Minmax-distance approximation and separation problems: geometrical properties

Plastria

Carrizosa

2010

Math. Program.

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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.

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Section: Theorem 2 There Always Exists a Constrained Center Hyperplanementioning

confidence: 52%

Section: Gauge Distances and Center Hyperplanesmentioning

confidence: 99%

“…sphere, hyperplane or polygon in the plane), leading to location models for extensive facilities. See [2,11,36] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Minmax-distance approximation and separation problems: geometrical properties

Plastria

Carrizosa

2010

Math. Program.

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show abstract

“…Extensive facility location concerns locating a dimensional structure, for example a path or a cycle on a network. This falls outside the scope of this paper -we refer to Mesa and Boffey (1996), Labbé et al (1988) and Díaz-Báñez et al (2004) for reviews.…”

Section: Vehicle Routing-allocation Problemsmentioning

confidence: 99%