Mathematical Programming

, Volume 6, Issue 1, pp 89–104

Local convergence in Fermat's problem

  • I. Norman Katz

DOI: 10.1007/BF01580224

Cite this article as:
Katz, I.N. Mathematical Programming (1974) 6: 89. doi:10.1007/BF01580224


The general Fermat problem is to find the minimum of the weighted sum of distances fromm destination points in Euclideann-space. Kuhn recently proved that a classical iterative algorithm converges to the unique minimizing point , for any choice of the initial point except for a denumerable set. In this note, it is shown that although convergence is global, the rapidity of convergence depends strongly upon whether or not  is a destination.

If  is not a destination, then locally convergence is always linear with upper and lower asymptotic convergence boundsλ andλ′ (λ ≥ 1/2, whenn=2). If  is a destination, then convergence can be either linear, quadratic or sublinear. Three numerical examples which illustrate the different possibilities are given and comparisons are made with the use of Steffensen's scheme to accelerate convergence.

Copyright information

© The Mathematical Programming Society 1974

Authors and Affiliations

  • I. Norman Katz
    • 1
  1. 1.Washington UniversitySt. LouisUSA