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 (unauthenticated)
  • Patricia Edith Alvarez Rodriguez 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
  • Heyssen Dueñes Chávez 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 http://orcid.org/0000-0001-5083-7800 (unauthenticated)

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.

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Published

2019-07-21

How to Cite

Weber’s problem on the Riemannian Manifolds: Some upper bounds for the minimun Weber’s function. (2019). Selecciones Matemáticas, 6(01), 108-118. https://doi.org/10.17268/sel.mat.2019.01.13

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