Open questions concerning Weiszfeld’s algorithm for the Fermat-Weber location problem

Open questions concerning Weiszfeld’s algorithm for the Fermat-Weber location problem

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Article ID: iaor1989664
Country: Netherlands
Volume: 44
Issue: 3
Start Page Number: 293
End Page Number: 295
Publication Date: Nov 1989
Journal: Mathematical Programming (Series A)
Authors: ,
Keywords: location
Abstract:

The Fermat-Weber location problem is to find a point in n that minimizes the sum of the weighted Euclidean distances from m given points in n. A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if the n 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. The authors demonstrate that Kuhn’s convergence theorem is not always correct. They then conjecture that if this algorithm is initiated at the affine subspace spanned by the m given points, the convergence is ensured for all but a denumerable number of starting points.

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