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Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

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  • Boris Mordukhovich

    (Wayne State University)

  • Nguyen Mau Nam

    (The University of Texas–Pan American)

Abstract

In this paper we develop new applications of variational analysis and generalized differentiation to the following optimization problem and its specifications: given n closed subsets of a Banach space, find such a point for which the sum of its distances to these sets is minimal. This problem can be viewed as an extension of the celebrated Fermat-Torricelli problem: given three points on the plane, find another point that minimizes the sum of its distances to the designated points. The generalized Fermat-Torricelli problem formulated and studied in this paper is of undoubted mathematical interest and is promising for various applications including those frequently arising in location science, optimal networks, etc. Based on advanced tools and recent results of variational analysis and generalized differentiation, we derive necessary as well as necessary and sufficient optimality conditions for the extended version of the Fermat-Torricelli problem under consideration, which allow us to completely solve it in some important settings. Furthermore, we develop and justify a numerical algorithm of the subgradient type to find optimal solutions in convex settings and provide its numerical implementations.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:3:d:10.1007_s10957-010-9761-7
    DOI: 10.1007/s10957-010-9761-7
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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. Boris Mordukhovich & Nguyen Nam, 2010. "Limiting subgradients of minimal time functions in Banach spaces," Journal of Global Optimization, Springer, vol. 46(4), pages 615-633, April.
    3. 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.
    4. 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.
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    Cited by:

    1. 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.
    2. Nguyen Mau Nam & Maria Cristina Villalobos & Nguyen Thai An, 2012. "Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 768-791, September.
    3. 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.

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