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Global Convergence of a Generalized Iterative Procedure for the Minisum Location Problem with lp Distances

Author

Listed:
  • Jack Brimberg

    (Royal Military College, Kingston, Ontario, Canada)

  • Robert F. Love

    (McMaster University, Hamilton, Ontario, Canada)

Abstract

This paper considers a general form of the single facility minisum location problem (also referred to as the Fermat-Weber problem), where distances are measured by an l p norm. An iterative solution algorithm is given which generalizes the well-known Weiszfeld procedure for Euclidean distances. Global convergence of the algorithm is proven for any value of the parameter p in the closed interval [1, 2], provided an iterate does not coincide with a singular point of the iteration functions. However, for p > 2, the descent property of the algorithm and as a result, global convergence, are no longer guaranteed. These results generalize the work of Kuhn for Euclidean ( p = 2) distances.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:oropre:v:41:y:1993:i:6:p:1153-1163
    DOI: 10.1287/opre.41.6.1153
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    Citations

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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. J. Brimberg & S. Salhi, 2005. "A Continuous Location-Allocation Problem with Zone-Dependent Fixed Cost," Annals of Operations Research, Springer, vol. 136(1), pages 99-115, April.
    4. Jack Brimberg & Pierre Hansen & Nenad Mladenović & Eric D. Taillard, 2000. "Improvements and Comparison of Heuristics for Solving the Uncapacitated Multisource Weber Problem," Operations Research, INFORMS, vol. 48(3), pages 444-460, June.
    5. 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.
    6. 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.
    7. Turkensteen, Marcel & Klose, Andreas, 2012. "Demand dispersion and logistics costs in one-to-many distribution systems," European Journal of Operational Research, Elsevier, vol. 223(2), pages 499-507.
    8. 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.
    9. Jack Brimberg & Henrik Juel & Anita Schöbel, 2002. "Linear Facility Location in Three Dimensions---Models and Solution Methods," Operations Research, INFORMS, vol. 50(6), pages 1050-1057, December.
    10. AltInel, I. Kuban & Durmaz, Engin & Aras, Necati & ÖzkIsacIk, Kerem Can, 2009. "A location-allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations," European Journal of Operational Research, Elsevier, vol. 198(3), pages 790-799, November.
    11. Hanif D. Sherali & Intesar Al-Loughani & Shivaram Subramanian, 2002. "Global Optimization Procedures for the Capacitated Euclidean and l p Distance Multifacility Location-Allocation Problems," Operations Research, INFORMS, vol. 50(3), pages 433-448, June.
    12. M. Hakan Akyüz & Temel Öncan & İ. Kuban Altınel, 2019. "Branch and bound algorithms for solving the multi-commodity capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 279(1), pages 1-42, August.
    13. Brimberg, Jack & Juel, Henrik, 1998. "A bicriteria model for locating a semi-desirable facility in the plane," European Journal of Operational Research, Elsevier, vol. 106(1), pages 144-151, April.
    14. Justo Puerto & Antonio Rodríguez-Chía, 2006. "New models for locating a moving service facility," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(1), pages 31-51, February.
    15. M. Akyüz & İ. Altınel & Temel Öncan, 2014. "Location and allocation based branch and bound algorithms for the capacitated multi-facility Weber problem," Annals of Operations Research, Springer, vol. 222(1), pages 45-71, November.

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