Abstract
The Fermat—Weber location problem requires finding a point in ℝN that minimizes the sum of weighted Euclidean distances tom given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfeld's algorithm was proven in a seminal paper by Kuhn in 1973. However, since them given points are singular points of the iteration functions, convergence is conditional on none of the iterates coinciding with one of the given points. In addressing this problem, Kuhn concluded that whenever them given points are not collinear, Weiszfeld's algorithm will converge to the unique optimal solution except for a denumerable set of starting points. As late as 1989, Chandrasekaran and Tamir demonstrated with counter-examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of ℝN. We resolve this open question by proving that Weiszfeld's algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimensionN.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Brimberg and R.F. Love, “Local convergence in a generalized Fermat—Weber problem,”Annals of Operations Research 40 (1992) 33–66.
J. Brimberg and R.F. Love, “Global convergence of a generalized iterative procedure for the minisum location problem withl p distances,”Operations Research 41 (1993) 1153–1163.
R. Chandrasekaran and A. Tamir, “Open questions concerning Weiszfeld's algorithm for the Fermat—Weber location problem,”Mathematical Programming 44 (1989) 293–295.
L. Cooper, “Location—allocation problems,”Operations Research 11 (1963) 37–52.
H. Juel and R.F. Love, “Fixed point optimality criteria for the location problem with arbitrary norms,”Journal of the Operational Research Society 32 (1981) 891–897.
I.N. Katz, “Local convergence in Fermat's problem,”Mathematical Programming 6 (1974) 89–104.
H.W. Kuhn, “A note on Fermat's problem,”Mathematical Programming 4 (1973) 98–107.
H.W. Kuhn and R.E. Kuenne, “An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics,”Journal of Regional Science 4 (1962) 21–34.
W. Miehle, “Link-length minimization in networks,”Operations Research 6 (1958) 232–243.
A.E. Taylor and W.R. Mann,Advanced Calculus (Xerox College Publishing, Lexington, MA, 2nd ed., 1972).
E. Weiszfeld, “Sur le point par lequel la somme des distances den points donnés est minimum,”Tohoku Mathematics Journal 43 (1937) 355–386.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brimberg, J. The Fermat—Weber location problem revisited. Mathematical Programming 71, 71–76 (1995). https://doi.org/10.1007/BF01592245
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01592245