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

@article{Chandrasekaran1989OpenQC, title={Open questions concerning Weiszfeld's algorithm for the Fermat-Weber location problem}, author={Ramaswamy Chandrasekaran and Arie Tamir}, journal={Mathematical Programming}, year={1989}, volume={44}, pages={293-295} }

The Fermat—Weber location problem is to find a point in ℝn that minimizes the sum of the weighted Euclidean distances fromm given points in ℝn. A popular iterative solution method for this problem was first introduced by Weiszfeld in 1937. In 1973 Kuhn claimed that if them given points are not collinear then for all but a denumerable number of starting points the sequence of iterates generated by Weiszfeld's scheme converges to the unique optimal solution. We demonstrate that Kuhn's convergence…

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## 98 Citations

The Fermat—Weber location problem revisited

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- 1995

It is proved that Weiszfeld's algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimensionN.

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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.

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- 2012

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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…

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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…

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## References

SHOWING 1-7 OF 7 REFERENCES

Local convergence in Fermat's problem

- Mathematics, Computer ScienceMath. 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

- Mathematics, Computer ScienceMath. 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.

A quadratically convergent method for minimizing a sum of euclidean norms

- Mathematics, Computer ScienceMath. Program.
- 1983

A new method is presented which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along which F is locally differentiable, and has quadratic convergence to a solutionx under given conditions.

A subgradient algorithm for certain minimax and minisum problems

- Mathematics, Computer ScienceMath. Program.
- 1978

Convergence of the algorithm to an epsilon-optimal solution is proven and its effectiveness is demonstrated by solving a number of location problems and linear approximation problems.

On Solving Multifacility Location Problems using a Hyperboloid Approximation Procedure

- Mathematics
- 1973

Abstract An iterative solution method is presented for solving multifacility location problems involving rectilinear and/or Euclidean distances. The iterative procedure is based on the use of an…

A second-order method for solving the continuous multifacility location problem

- Mathematics
- 1982

A unified and numerically stable second-order approach to the continuous multifacility location problem is presented. Although details are initially given for only the unconstrained Euclidean norm…

Time Bounds for Selection

- Computer Science, MathematicsJ. Comput. Syst. Sci.
- 1973

The number of comparisons required to select the i-th smallest of n numbers is shown to be at most a linear function of n by analysis of a new selection algorithm-PICK. Specifically, no more than…