Weiszfeld’s Method: Old and New Results

Amir Beck*, Shoham Sabach

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In 1937, the 16-years-old Hungarian mathematician Endre Weiszfeld, in a seminal paper, devised a method for solving the Fermat–Weber location problem—a problem whose origins can be traced back to the seventeenth century. Weiszfeld’s method stirred up an enormous amount of research in the optimization and location communities, and is also being discussed and used till these days. In this paper, we review both the past and the ongoing research on Weiszfed’s method. The existing results are presented in a self-contained and concise manner—some are derived by new and simplified techniques. We also establish two new results using modern tools of optimization. First, we establish a non-asymptotic sublinear rate of convergence of Weiszfeld’s method, and second, using an exact smoothing technique, we present a modification of the method with a proven better rate of convergence.

Original languageEnglish
Pages (from-to)1-40
Number of pages40
JournalJournal of Optimization Theory and Applications
Issue number1
StatePublished - Jan 2014
Externally publishedYes


  • Complexity analysis
  • Fermat–Weber problem
  • Gradient method
  • Localization theory
  • Weiszfeld’s method


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