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A proximal method for solving nonlinear minmax location problems with perturbed minimal time functions via conjugate duality

Author

Listed:
  • Sorin-Mihai Grad

    (University of Vienna)

  • Oleg Wilfer

    (Chemnitz University of Technology)

Abstract

We investigate via a conjugate duality approach general nonlinear minmax location problems formulated by means of an extended perturbed minimal time function, necessary and sufficient optimality conditions being delivered together with characterizations of the optimal solutions in some particular instances. A parallel splitting proximal point method is employed in order to numerically solve such problems and their duals. We present the computational results obtained in matlab on concrete examples, successfully comparing these, where possible, with earlier similar methods from the literature. Moreover, the dual employment of the proximal method turns out to deliver the optimal solution to the considered primal problem faster than the direct usage on the latter. Since our technique successfully solves location optimization problems with large data sets in high dimensions, we envision its future usage on big data problems arising in machine learning.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:74:y:2019:i:1:d:10.1007_s10898-019-00746-5
    DOI: 10.1007/s10898-019-00746-5
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    References listed on IDEAS

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    1. Nguyen Mau Nam & Nguyen Hoang & Nguyen Thai An, 2014. "Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 483-509, February.
    2. 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.
    3. Gert Wanka & Oleg Wilfer, 2017. "A Lagrange duality approach for multi-composed optimization problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(2), pages 288-313, July.
    4. 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.
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