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Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design

Author

Listed:
  • Jean-Thomas Camino

    (Université de Toulouse, CNRS, UPS
    Airbus Defence and Space, Space Systems, Telecommunication Systems Department)

  • Christian Artigues

    (Université de Toulouse, CNRS, UPS)

  • Laurent Houssin

    (Université de Toulouse, CNRS, UPS)

  • Stéphane Mourgues

    (Airbus Defence and Space, Space Systems, Telecommunication Systems Department)

Abstract

Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k-center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.

Suggested Citation

  • Jean-Thomas Camino & Christian Artigues & Laurent Houssin & Stéphane Mourgues, 2019. "Linearization of Euclidean norm dependent inequalities applied to multibeam satellites design," Computational Optimization and Applications, Springer, vol. 73(2), pages 679-705, June.
  • Handle: RePEc:spr:coopap:v:73:y:2019:i:2:d:10.1007_s10589-019-00083-z
    DOI: 10.1007/s10589-019-00083-z
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    References listed on IDEAS

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    1. Hatem Fayed & Amir Atiya, 2013. "Erratum to: A mixed breadth-depth first strategy for the branch and bound tree of Euclidean k-center problems," Computational Optimization and Applications, Springer, vol. 54(3), pages 705-705, April.
    2. Kata Kiatmanaroj & Christian Artigues & Laurent Houssin & Frédéric Messine, 2013. "Frequency assignment in a SDMA satellite communication system with beam decentring feature," Computational Optimization and Applications, Springer, vol. 56(2), pages 439-455, October.
    3. Hatem Fayed & Amir Atiya, 2013. "A mixed breadth-depth first strategy for the branch and bound tree of Euclidean k-center problems," Computational Optimization and Applications, Springer, vol. 54(3), pages 675-703, April.
    4. A. Sutou & Y. Dai, 2002. "Global Optimization Approach to Unequal Global Optimization Approach to Unequal Sphere Packing Problems in 3D," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 671-694, September.
    5. Harman, Radoslav & Lacko, Vladimír, 2010. "On decompositional algorithms for uniform sampling from n-spheres and n-balls," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2297-2304, November.
    6. Aharon Ben-Tal & Arkadi Nemirovski, 2001. "On Polyhedral Approximations of the Second-Order Cone," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 193-205, May.
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    Cited by:

    1. Huiliang Liu & Yao Chu & Yulong Zhang & Weiguo Hou & Yinqiao Li & Yuan Yao & Yaxing Cai, 2021. "Strategy of Multi-Beam Spot Allocation for GEO Data Relay Satellite Based on Modified K-Means Algorithm," Mathematics, MDPI, vol. 9(15), pages 1-16, July.
    2. Aloïs Duguet & Christian Artigues & Laurent Houssin & Sandra Ulrich Ngueveu, 2022. "Properties, Extensions and Application of Piecewise Linearization for Euclidean Norm Optimization in $$\mathbb {R}^2$$ R 2," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 418-448, November.

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