Cotas superiores para el problema de Weber sobre una variedad Riemanniana N-Dimensional
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Date
2025
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Universidad Nacional de Trujillo
Abstract
El problema de Weber es un problema clásico en teoría de localización de servicios; el cual consiste en determinar una ubicación que permita minimizar la suma de distancias ponderadas a “m” lugares fijados previamente.
Matemáticamente, este problema consiste en hallar un punto en𝑅𝑛, que minimice la suma de distancias Euclideanas ponderadas a “m” punto fijados previamente.
Weiszfeld en 1937, formuló un algoritmo para determinar mediante un proceso iterativo una sucesión de puntos en 𝑅𝑛; sucesión que era convergente al punto que minimizaba la función de Weber: función definida por la suma de las distancias Euclideanas ponderadas.
En 1978, Drezner y Wesolowski, generalizaron el problema de Weber a la superficie de la esfera unitaria; dado que la superficie de la esfera se usaba para modelar al planeta tierra; pues cuando se consideraban regiones extensas sobre la superficie terrestre, la geometría del plano no proporcionaba buenos resultados.
En el 2009, Fletcher et al, generaliza el problema de Weber a variedades Riemannianas de dimensión finita, y prueba que este problema definido sobre un conjunto fuertemente convexo, tiene una única solución; siempre y cuando la curvatura seccional sea positiva y acotada.
En este trabajo de investigación, se considera el problema de Weber definido sobre una bola fuertemente convexa en una variedad Riemanniana n-dimensional de clase 𝐶∞, y se determinan cotas superiores para la función de Weber definida sobre la bola fuertemente convexa, bajo la condición de que la curvatura seccional de la variedad Riemanniana sea positiva y acotada.
The Weber problem is a classic problem in service location theory; it consists of determining a location that allows to minimize the sum of weighted distances to “m” previously fixed places. Mathematically, this problem consists of finding a point in 𝑅𝑛, which minimizes the sum of weighted Euclidean distances to “m” previously fixed points. In 1937, Weiszfeld formulated an algorithm to determine, through an iterative process, a succession of points in 𝑅𝑛; a succession that was convergent to the point that minimized the Weber function: function defined by the sum of the weighted Euclidean distances. In 1978, Drezner and Wesolowski generalized the Weber problem to the surface of the unitary sphere; given that the surface of the sphere was used to model the planet Earth; when extensive regions on the Earth's surface were considered, the geometry of the plane did not provide good results. In 2009, Fletcher et al. generalized the Weber problem to finite-dimensional Riemannian manifolds, and proved that this problem, defined on a strongly convex set, has a unique solution; provided that the sectional curvature is positive and bounded. In this research work, we consider the Weber problem defined on a strongly convex ball on an n-dimensional Riemannian manifold of class 𝐶∞, and determine upper bounds for the Weber function defined on the strongly convex ball, under the condition that the sectional curvature of the Riemannian manifold is positive and bounded.
The Weber problem is a classic problem in service location theory; it consists of determining a location that allows to minimize the sum of weighted distances to “m” previously fixed places. Mathematically, this problem consists of finding a point in 𝑅𝑛, which minimizes the sum of weighted Euclidean distances to “m” previously fixed points. In 1937, Weiszfeld formulated an algorithm to determine, through an iterative process, a succession of points in 𝑅𝑛; a succession that was convergent to the point that minimized the Weber function: function defined by the sum of the weighted Euclidean distances. In 1978, Drezner and Wesolowski generalized the Weber problem to the surface of the unitary sphere; given that the surface of the sphere was used to model the planet Earth; when extensive regions on the Earth's surface were considered, the geometry of the plane did not provide good results. In 2009, Fletcher et al. generalized the Weber problem to finite-dimensional Riemannian manifolds, and proved that this problem, defined on a strongly convex set, has a unique solution; provided that the sectional curvature is positive and bounded. In this research work, we consider the Weber problem defined on a strongly convex ball on an n-dimensional Riemannian manifold of class 𝐶∞, and determine upper bounds for the Weber function defined on the strongly convex ball, under the condition that the sectional curvature of the Riemannian manifold is positive and bounded.
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Keywords
Problema de Weber, Variedad Riemanniana, Mediana geométrica ponderada, Conjunto fuertemente convexo