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The Generalized Fermat–Torricelli Problem in Hilbert Spaces

Author

Listed:
  • Simeon Reich

    (The Technion – Israel Institute of Technology)

  • Truong Minh Tuyen

    (Thai Nguyen University of Sciences)

Abstract

We study the generalized Fermat–Torricelli problem and the split feasibility problem with multiple output sets in Hilbert spaces. We first introduce the generalized Fermat–Torricelli problem, and propose and analyze a subgradient algorithm for solving this model problem. Then we study the convergence of variants of our proposed algorithm for solving the split feasibility problem with multiple output sets. Our algorithms for solving this problem are completely different from previous ones because we do not use the least squares sum method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:196:y:2023:i:1:d:10.1007_s10957-022-02113-z
    DOI: 10.1007/s10957-022-02113-z
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    References listed on IDEAS

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    1. H. Martini & K.J. Swanepoel & G. Weiss, 2002. "The Fermat–Torricelli Problem in Normed Planes and Spaces," Journal of Optimization Theory and Applications, Springer, vol. 115(2), pages 283-314, November.
    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. Simeon Reich & Truong Minh Tuyen & Mai Thi Ngoc Ha, 2021. "An optimization approach to solving the split feasibility problem in Hilbert spaces," Journal of Global Optimization, Springer, vol. 79(4), pages 837-852, April.
    5. 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.
    6. T. V. Tan, 2010. "An Extension of the Fermat-Torricelli Problem," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 735-744, September.
    7. 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.
    8. Boris Mordukhovich & Nguyen Mau Nam, 2011. "Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 431-454, March.
    9. William Miehle, 1958. "Link-Length Minimization in Networks," Operations Research, INFORMS, vol. 6(2), pages 232-243, April.
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