# The Fermat—Weber location problem revisited

@article{Brimberg1995TheFL, title={The Fermat—Weber location problem revisited}, author={Jack Brimberg}, journal={Mathematical Programming}, year={1995}, volume={71}, pages={71-76} }

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…

## 79 Citations

Constrained Fermat-Torricelli-Weber Problem in real Hilbert Spaces

- Mathematics
- 2018

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

- Mathematics
- 2012

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

- EconomicsAdv. Oper. Res.
- 2011

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 projected Weiszfeld algorithm for the box-constrained Weber location problem

- MathematicsAppl. Math. Comput.
- 2011

A geometric perspective of the Weiszfeld algorithm for solving the Fermat-Weber problem

- MathematicsRAIRO Oper. Res.
- 2016

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

- Mathematics
- 2009

The Fermat-Weber problem consists in finding a point in R that minimizes the weighted sum of distances from m points in R that are not collinear. An application that motivated this problem is the…

The Fermat-Torricelli problem and Weiszfeld’s algorithm in the light of convex analysis

- MathematicsJournal of Applied and Numerical Optimization
- 2019

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

- Computer ScienceComput. 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.

FURTHER NOTES ON CONVERGENCE OF THE WEISZFELD ALGORITHM

- Mathematics
- 2003

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…

## References

SHOWING 1-10 OF 10 REFERENCES

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

- MathematicsMath. Program.
- 1989

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

- MathematicsAnn. Oper. Res.
- 1992

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

- MathematicsMath. 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

- MathematicsMath. 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

- MathematicsOper. Res.
- 1993

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

- Mathematics
- 1981

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.

Link-Length Minimization in Networks

- Computer Science
- 1958

Three general methods are described for attacking link-length minimization problems. All methods assume a system of fixed points that are to be interconnected by a network of links the sum of whose…

Location-Allocation Problems

- Business
- 1963

The calculational aspects of solving certain classes of location-allocation problems are presented. Both exact extremal equations and a heuristic method are presented for solving these problems.…

AN EFFICIENT ALGORITHM FOR THE NUMERICAL SOLUTION OF THE GENERALIZED WEBER PROBLEM IN SPATIAL ECONOMICS

- Economics
- 1962

Sur le point par lequel la somme des distances de n points donn6s est minimum

- Tohoku Mathematics Journal 43 (1937) 355-386.
- 1937