Abstract
The Fermat—Weber location problem is to find a point in ℝn that minimizes the sum of the weighted Euclidean distances fromm given points in ℝn. A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence theorem is not always correct. We then conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Blum, R.W. Floyd, V.R. Pratt, R.L. Rivest and R.E. Tarjan, “Time bounds for selection,”Journal of Computer and System Sciences 7 (1972) 448–461.
P.H. Calamai and A.R. Conn, “A second-order method for solving the continuous multifacility location problem,” in: G.A. Watson, ed.,Numerical analysis: Proceedings of the Ninth Biennial Conference, Dundee, Scotland, Lecture Notes in Mathematics, Vol. 912 (Springer, Berlin, Heidelberg and New York, 1982) pp. 1–25.
J.A. Chatelon, D.W. Hearn and T.J. Lowe, “A subgradient algorithm for certain minimax and minisum problems,”Mathematical Programming 15 (1978) 130–145.
J.W. Eyster, J.A. White and W.W. Wierwille, “On solving multifacility location problems using a hyperboloid approximation procedure,”AIEE Transactions 5 (1973) 1–6.
I.N. Katz, “Local convergence in Fermat's problem,”Mathematical Programming 6 (1974) 89–104.
H.W. Kuhn, “On a pair of dual nonlinear programs,” in: J. Abadie, ed.,Nonlinear Programming (North-Holland, Amsterdam, 1967) pp. 38–54.
H.W. Kuhn, “A note on Fermat's problem,”Mathematical Programming 4 (1973) 98–107.
M.L. Overton, “A quadratically convergence method for minimizing a sum of Euclidean norms,”Mathematical Programming 27 (1983) 34–63.
A. Weber,Theory of the Location of Industries, translated by Carl J. Friedrich (The University of Chicago Press, Chicago, 1937).
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
Chandrasekaran, R., Tamir, A. Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem. Mathematical Programming 44, 293–295 (1989). https://doi.org/10.1007/BF01587094
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01587094