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Weiszfeld’s Method: Old and New Results

Author

Listed:
  • Amir Beck

    (Technion—Israel Institute of Technology)

  • Shoham Sabach

    (Tel Aviv University)

Abstract

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.

Suggested Citation

  • Amir Beck & Shoham Sabach, 2015. "Weiszfeld’s Method: Old and New Results," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 1-40, January.
    • Handle: RePEc:spr:joptap:v:164:y:2015:i:1:d:10.1007_s10957-014-0586-7
    DOI: 10.1007/s10957-014-0586-7
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    References listed on IDEAS

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    1. Yaakov S. Kupitz & Horst Martini & Margarita Spirova, 2013. "The Fermat–Torricelli Problem, Part I: A Discrete Gradient-Method Approach," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 305-327, August.
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