A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem

Abstract

The Generalized Fermat Problem (in the plane) is: given n ≥ 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 ̄ ∗ 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.