Weber’s problem on the Riemannian Manifolds: Some upper bounds for the minimun Weber’s function

Authors

  • Franco Rubio López Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Ciudad Universitaria, Trujillo-Perú http://orcid.org/0000-0002-0168-3806
  • Patricia Edith Alvarez Rodriguez Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Ciudad Universitaria, Trujillo-Perú
  • Heyssen Dueñes Chávez Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Ciudad Universitaria, Trujillo-Perú http://orcid.org/0000-0001-5083-7800

DOI:

https://doi.org/10.17268/sel.mat.2019.01.13

Keywords:

The Weber problema, Weighted Geometric Median, Riemannian manifold, Strongly convex set

Abstract

In this paper we obtain some upper bounds for the minimum of the Weber function on a strongly convex ball in a Riemannian manifold with positive sectional curvature; where the minimum is reached on the weighted geometric median of “m” given points in the strongly convex.

Author Biography

Franco Rubio López, Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Ciudad Universitaria, Trujillo-Perú

Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Ciudad Universitaria, Trujillo-Perú

References

Aftab, K., Hartley, R., and Trumpf, J. Generalized Weiszfeld Algorithms for Lq Optimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 37(4), 728– 745, 2015. doi:10.1109/tpami.2014.2353625

Drezner, W and Wesolowsky, G.O. Facility Location on the Sphere. Journal of the Operational Research Society, 29, 997-1004, 1978.

Drezner, W. A Solution to the Weber Location Problem on the Sphere. Journal of the Operational Research Society, 36, 333-338, 1985.

Fletcher, T; Venkatasubramanian, V and Joshi, S. The geometric median on Riemannian manifolds with application to robust atlas estimation. NeuroImage 45, s143-s152, 2009.

Hansen, P; Jaumard, B and Krau, S. A algorithm for Weber’s Problem on the Sphere. Location Science 3(4), 217-237, 1995.

P. Do Carmo, M. Geometria Riemanniana. IMPA, Rio de Janeiro, 1979.

Weiszfeld, E. V. Sur le point pour lequel la Somme des distances de n point donnés est minisum. The Tohoku Mathematical Journal, 43, 335-386, 1937.

Wendel, R and Hurter, A. Location Theory, dominance and convexity. Operations Research, 21(1), 314-320, 1973.

Wesolowsky, G.O. Location Problem on a Sphere. Regional Science and Urban Economics, 12, 495-508, 1982.

Downloads

Published

2019-07-21

How to Cite

Rubio López, F., Alvarez Rodriguez, P. E., & Dueñes Chávez, H. (2019). Weber’s problem on the Riemannian Manifolds: Some upper bounds for the minimun Weber’s function. Selecciones Matemáticas, 6(01), 108-118. https://doi.org/10.17268/sel.mat.2019.01.13