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A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem. (English) Zbl 1189.90086
Summary: The Generalized Fermat Problem (in the plane) is: given $n\ge 3$ 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:
 90B85 Continuous location 52B55 Computational aspects related to geometric convexity