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On the Convergence of a Class of Iterative Methods for Solving the Weber Location Problem

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  • Lawrence M. Ostresh

    (University of Wyoming, Laramie, Wyoming)

Abstract

The location problem is to find a point M whose sum of weighted distances from m vertices in p -dimensional Euclidean space is a minimum. The best-known algorithm for solving the location problem is an iterative scheme devised by Weiszfeld in 1937. The procedure will not converge if some nonoptimal vertex is an iterate, however. This paper solves the problem of vertex iterates and presents a general proof permitting a variable step length (within certain bounds). This property is used, in particular, to show the convergence of a modified gradient Newton-Raphson type of procedure.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:26:y:1978:i:4:p:597-609
    DOI: 10.1287/opre.26.4.597
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    Cited by:

    1. Zvi Drezner, 2009. "On the convergence of the generalized Weiszfeld algorithm," Annals of Operations Research, Springer, vol. 167(1), pages 327-336, March.
    2. Victor Blanco & Justo Puerto & Safae El Haj Ben Ali, 2014. "Revisiting several problems and algorithms in continuous location with $$\ell _\tau $$ ℓ τ norms," Computational Optimization and Applications, Springer, vol. 58(3), pages 563-595, July.
    3. Venkateshan, Prahalad & Ballou, Ronald H. & Mathur, Kamlesh & Maruthasalam, Arulanantha P.P., 2017. "A Two-echelon joint continuous-discrete location model," European Journal of Operational Research, Elsevier, vol. 262(3), pages 1028-1039.
    4. Murray, Alan T. & Church, Richard L. & Feng, Xin, 2020. "Single facility siting involving allocation decisions," European Journal of Operational Research, Elsevier, vol. 284(3), pages 834-846.
    5. Jianlin Jiang & Xiaoming Yuan, 2012. "A Barzilai-Borwein-based heuristic algorithm for locating multiple facilities with regional demand," Computational Optimization and Applications, Springer, vol. 51(3), pages 1275-1295, April.
    6. Jianlin Jiang & Su Zhang & Yibing Lv & Xin Du & Ziwei Yan, 2020. "An ADMM-based location–allocation algorithm for nonconvex constrained multi-source Weber problem under gauge," Journal of Global Optimization, Springer, vol. 76(4), pages 793-818, April.
    7. Adi Ben-Israel & Cem Iyigun, 2008. "Probabilistic D-Clustering," Journal of Classification, Springer;The Classification Society, vol. 25(1), pages 5-26, June.
    8. 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.
    9. Jiang, Jian-Lin & Yuan, Xiao-Ming, 2008. "A heuristic algorithm for constrained multi-source Weber problem - The variational inequality approach," European Journal of Operational Research, Elsevier, vol. 187(2), pages 357-370, June.
    10. Richard L. Church & Zvi Drezner & Pawel Kalczynski, 2023. "Extensions to the planar p-median problem," Annals of Operations Research, Springer, vol. 326(1), pages 115-135, July.
    11. Burkard, Rainer E. & Galavii, Mohammadreza & Gassner, Elisabeth, 2010. "The inverse Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 206(1), pages 11-17, October.
    12. Zvi Drezner & Siegfried Schaible & David Simchi‐Levi, 1990. "Queueing‐location problems on the plane," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(6), pages 929-935, December.

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