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On the convergence of the generalized Weiszfeld algorithm

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  • Zvi Drezner

Abstract

In this paper we consider Weber-like location problems. The objective function is a sum of terms, each a function of the Euclidean distance from a demand point. We prove that a Weiszfeld-like iterative procedure for the solution of such problems converges to a local minimum (or a saddle point) when three conditions are met. Many location problems can be solved by the generalized Weiszfeld algorithm. There are many problem instances for which convergence is observed empirically. The proof in this paper shows that many of these algorithms indeed converge. Copyright Springer Science+Business Media, LLC 2009

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  • Zvi Drezner, 2009. "On the convergence of the generalized Weiszfeld algorithm," Annals of Operations Research, Springer, vol. 167(1), pages 327-336, March.
    • Handle: RePEc:spr:annopr:v:167:y:2009:i:1:p:327-336:10.1007/s10479-008-0336-z
    DOI: 10.1007/s10479-008-0336-z
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    References listed on IDEAS

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    1. Justo Puerto & Antonio M. Rodríguez-Chía, 1999. "Location of a moving service facility," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(3), pages 373-393, July.
    2. Jack Brimberg & Robert F. Love, 1993. "Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances," Operations Research, INFORMS, vol. 41(6), pages 1153-1163, December.
    3. 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.
    4. Tammy Drezner & Zvi Drezner, 1997. "Replacing continuous demand with discrete demand in a competitive location model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 44(1), pages 81-95, February.
    5. Tammy Drezner & Zvi Drezner, 2004. "Finding the optimal solution to the Huff based competitive location model," Computational Management Science, Springer, vol. 1(2), pages 193-208, July.
    6. 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:

    1. Drezner, Zvi & Eiselt, H.A., 2024. "Competitive location models: A review," European Journal of Operational Research, Elsevier, vol. 316(1), pages 5-18.
    2. Nguyen Mau Nam & R. Blake Rector & Daniel Giles, 2017. "Minimizing Differences of Convex Functions with Applications to Facility Location and Clustering," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 255-278, April.
    3. Zvi Drezner & Carlton Scott, 2013. "Location of a distribution center for a perishable product," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(3), pages 301-314, December.
    4. Rodríguez-Chía, Antonio M. & Espejo, Inmaculada & Drezner, Zvi, 2010. "On solving the planar k-centrum problem with Euclidean distances," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1169-1186, December.

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