The Fermat—Weber location problem revisited

  title={The Fermat—Weber location problem revisited},
  author={Jack Brimberg},
  journal={Mathematical Programming},
  • J. Brimberg
  • Published 30 November 1995
  • Mathematics
  • Mathematical Programming
The Fermat—Weber location problem requires finding a point in ℝN that minimizes the sum of weighted Euclidean distances tom given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfeld's algorithm was proven in a seminal paper by Kuhn in 1973. However, since them given points are singular points of the iteration… 
Constrained Fermat-Torricelli-Weber Problem in real Hilbert Spaces
The Fermat-Weber location problem requires finding a point in $\mathbb{R}^n$ that minimizes the sum of weighted Euclidean distances to $m$ given points. An iterative solution method for this problem
A Weiszfeld-like algorithm for a Weber location problem constrained to a closed and convex set
The Weber problem consists of finding a point in $\mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $\mathbbm{R}^n$ that are not collinear. An application that motivated
Noniterative Solution of Some Fermat-Weber Location Problems
This work describes a noniterative direct alternative to the iterative process of solving Fermat-Weber problems, based on the insight that the gradient components of the individual demand points can be considered as pooling forces with respect to the solution point.
A geometric perspective of the Weiszfeld algorithm for solving the Fermat-Weber problem
A geometric interpretation of the local convergence of the Fermat−Weber problem for the particular case of three points, with the solution constrained to be an interior point, which is fundamental to the present geometric interpretation.
SERIE “ A ” TRABAJOS DE MATEMÁTICA N o 95 / 09 A projected Weiszfeld ’ s algorithm for the box-constrained Weber location problem
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The Fermat-Torricelli problem and Weiszfeld’s algorithm in the light of convex analysis
In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is
A Newton Acceleration of the Weiszfeld Algorithm for Minimizing the Sum of Euclidean Distances
  • Yuying Li
  • Computer Science
    Comput. Optim. Appl.
  • 1998
A Newton algorithm with similar simplicity is proposed to solve a continuous multifacility location problem with the Euclidean distance measure and is proven to be globally convergent under similar assumptions for the Weiszfeld algorithm.
The Fermat-Weber problem is one of the most widely studied problems in classical location theory. In his previous work, Brimberg (1995) attempts to resolve a conjecture posed by Chandrasekaran and


Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem
It is demonstrated that Kuhn's convergence theorem is not always correct and it is conjecture that if this algorithm is initiated at the affine subspace spanned by them given points, the convergence is ensured for all but a denumerable number of starting points.
Local convergence in a generalized Fermat-Weber problem
A generalized version of the Fermat-Weber problem, where distances are measured by anlp norm and the parameterp takes on a value in the closed interval, which permits the choice of a continuum of distance measures from rectangular to Euclidean.
Local convergence in Fermat's problem
  • I. Katz
  • Mathematics
    Math. Program.
  • 1974
It is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination, and locally convergence is always linear with upper and lower asymptotic convergence boundsλ andλ′.
A note on Fermat's problem
  • H. Kuhn
  • Mathematics
    Math. Program.
  • 1973
This note calls attention to the work of Weiszfeld in 1937, who may have been the first to propose an iterative algorithm for the General Fermat Problem.
Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances
An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances, and global convergence of the algorithm is proven for any value of the parameter p in the closed interval.
Fixed Point Optimality Criteria for the Location Problem with Arbitrary Norms
A necessary and sufficient condition for the location of an existing facility to be the optimal location of the new facility is developed and some computational examples using the condition are given.
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Sur le point par lequel la somme des distances de n points donn6s est minimum
  • Tohoku Mathematics Journal 43 (1937) 355-386.
  • 1937