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On the convergence of the Weiszfeld algorithm. (English) Zbl 1065.90054
The Weiszfeld algorithm is an iterative algorithm to solve the Fermat-Weber problem. R. Chandrasekaran and A. Tamir [Math. Program., Ser. A 44, No. 3, 293–295 (1989; Zbl 0683.90026)] stated the following conjecture: If the convex hull of the set of vertices is of full dimension, then the set of initial points for which the sequence generated by the Weiszfeld algorithm yields in a vertex is denumerable. J. Brimberg [Math. Program. 71, No. 1 (A), 71–76 (1995; Zbl 0855.90075)] claimed to prove the conjecture and extends it to a necessary and sufficient condition. The authors show in this paper that Brimberg’s proof is not correct. Moreover, they show by examples that the conjecture cannot be extended to a necessary and sufficient condition.
MSC:
90B85Continuous location
90C30Nonlinear programming