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Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics

Author

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  • Nguyen Mau Nam

    (University of Texas-Pan American)

  • Maria Cristina Villalobos

    (University of Texas-Pan American)

  • Nguyen Thai An

    (Hue University)

Abstract

The smallest enclosing circle problem introduced in the nineteenth century by Sylvester asks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of this problem, called the smallest intersecting ball problem, was also considered recently: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper, we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to further extensions of the smallest enclosing circle problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:3:d:10.1007_s10957-012-0048-z
    DOI: 10.1007/s10957-012-0048-z
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    References listed on IDEAS

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    1. Nimrod Megiddo, 1983. "The Weighted Euclidean 1-Center Problem," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 498-504, 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. 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.
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    Cited by:

    1. Vo Si Trong Long, 2022. "An Invariant-Point Theorem in Banach Space with Applications to Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 440-464, August.

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