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Zbl 1189.90086
Katz, I.Norman; Vogl, Steven R.
A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem. (English)
[J] Comput. Math. Appl. 59, No. 1, 399-410 (2010). ISSN 0898-1221

Summary: The Generalized Fermat Problem (in the plane) is: given $n\geq 3$ destination points find the point $x^*$ which minimizes the sum of Euclidean distances from $x^*$ to each of the destination points. The Weiszfeld iterative algorithm for this problem is globally convergent, independent of the initial guess. Also, a test is available, a priori, to determine when $x^*$ is a destination point. This paper generalizes earlier work by the first author by introducing an asymmetric Euclidean distance in which, at each destination, the $x$-component is weighted differently from the $y$-component. A Weiszfeld algorithm is studied to compute $x^*$ and is shown to be a descent method which is globally convergent (except possibly for a denumerable number of starting points). Local convergence properties are characterized. When $x^*$ is not a destination point, the iteration matrix at $x^*$ is shown to be convergent and local convergence is always linear. When $x^*$ is a destination point, local convergence can be linear, sub-linear or super-linear, depending upon a computable criterion. A test, which does not require iteration, for $x^*$ to be a destination, is derived. Comparisons are made between the symmetric and asymmetric problems. Numerical examples are given.

MSC 2000:: *90B85 Continuous location
52B55 Computational aspects related to geometric convexity

Keywords: generalized Fermat location; asymmetric distance; Weiszfeld algorithm

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