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Duality results for nonlinear single minimax location problems via multi-composed optimization

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  • Gert Wanka

    (Chemnitz University of Technology)

  • Oleg Wilfer

    (Chemnitz University of Technology)

Abstract

In the framework of conjugate duality we discuss nonlinear and linear single minimax location problems with geometric constraints, where the gauges are defined by convex sets of a Fréchet space. The version of the nonlinear location problem is additionally considered with set-up costs. Associated dual problems for this kind of location problems will be formulated as well as corresponding duality statements. As conclusion of this paper, we give a geometrical interpretation of the optimal solutions of the dual problem of an unconstraint linear single minimax location problem when the gauges are a norm. For an illustration, an example in the Euclidean space will follow.

Suggested Citation

  • Gert Wanka & Oleg Wilfer, 2017. "Duality results for nonlinear single minimax location problems via multi-composed optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 401-439, October.
    • Handle: RePEc:spr:mathme:v:86:y:2017:i:2:d:10.1007_s00186-017-0603-3
    DOI: 10.1007/s00186-017-0603-3
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    References listed on IDEAS

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    1. Durier, Roland & Michelot, Christian, 1985. "Geometrical properties of the Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 20(3), pages 332-343, June.
    2. C. Michelot & F. Plastria, 2002. "An Extended Multifacility Minimax Location Problem Revisited," Annals of Operations Research, Springer, vol. 111(1), pages 167-179, March.
    3. Richard L. Francis, 1967. "Letter to the Editor—Some Aspects of a Minimax Location Problem," Operations Research, INFORMS, vol. 15(6), pages 1163-1169, December.
    4. Henrik Juel & Robert F. Love, 1981. "On the Dual of the Linearly Constrained Multifacility Location Problem with Arbitrary Norms," Transportation Science, INFORMS, vol. 15(4), pages 329-337, November.
    5. Y. Hinojosa & J. Puerto, 2003. "Single facility location problems with unbounded unit balls," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(1), pages 87-104, September.
    6. Wanka, Gert & Bot, Radu Ioan & Vargyas, Emese, 2007. "Duality for location problems with unbounded unit balls," European Journal of Operational Research, Elsevier, vol. 179(3), pages 1252-1265, June.
    7. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, November.
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    Cited by:

    1. Sorin-Mihai Grad & Oleg Wilfer, 2019. "A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality," Journal of Global Optimization, Springer, vol. 74(1), pages 121-160, May.
    2. Soumen Kumar Das & Sankar Kumar Roy & Gerhard Wilhelm Weber, 2020. "Heuristic approaches for solid transportation-p-facility location problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(3), pages 939-961, September.

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