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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. H. A. Eiselt & Vladimir Marianov, 2011. "Pioneering Developments in Location Analysis," International Series in Operations Research & Management Science, in: H. A. Eiselt & Vladimir Marianov (ed.), Foundations of Location Analysis, chapter 0, pages 3-22, Springer.
    2. 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.
    3. Leon Cooper, 1963. "Location-Allocation Problems," Operations Research, INFORMS, vol. 11(3), pages 331-343, June.
    4. E. Weiszfeld & Frank Plastria, 2009. "On the point for which the sum of the distances to n given points is minimum," Annals of Operations Research, Springer, vol. 167(1), pages 7-41, March.
    5. Lawrence M. Ostresh, 1978. "On the Convergence of a Class of Iterative Methods for Solving the Weber Location Problem," Operations Research, INFORMS, vol. 26(4), pages 597-609, August.
    6. William Miehle, 1958. "Link-Length Minimization in Networks," Operations Research, INFORMS, vol. 6(2), pages 232-243, April.
    7. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. James G. Morris, 1981. "Convergence of the Weiszfeld Algorithm for Weber Problems Using a Generalized “Distance” Function," Operations Research, INFORMS, vol. 29(1), pages 37-48, February.
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    Cited by:

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    3. Simeon Reich & Truong Minh Tuyen, 2023. "The Generalized Fermat–Torricelli Problem in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 78-97, January.

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