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Solving continuous set covering problems by means of semi-infinite optimization

Author

Listed:
  • Helene Krieg

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Tobias Seidel

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Jan Schwientek

    (Fraunhofer Institute for Industrial Mathematics ITWM)

  • Karl-Heinz Küfer

    (Fraunhofer Institute for Industrial Mathematics ITWM)

Abstract

This article introduces the new class of continuous set covering problems. These optimization problems result, among others, from product portfolio design tasks with products depending continuously on design parameters and the requirement that the product portfolio satisfies customer specifications that are provided as a compact set. We show that the problem can be formulated as semi-infinite optimization problem (SIP). Yet, the inherent non-smoothness of the semi-infinite constraint function hinders the straightforward application of standard methods from semi-infinite programming. We suggest an algorithm combining adaptive discretization of the infinite index set and replacement of the non-smooth constraint function by a two-parametric smoothing function. Under few requirements, the algorithm converges and the distance of a current iterate can be bounded in terms of the discretization and smoothing error. By means of a numerical example from product portfolio optimization, we demonstrate that the proposed algorithm only needs relatively few discretization points and thus keeps the problem dimensions small.

Suggested Citation

  • Helene Krieg & Tobias Seidel & Jan Schwientek & Karl-Heinz Küfer, 2022. "Solving continuous set covering problems by means of semi-infinite optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(1), pages 39-82, August.
    • Handle: RePEc:spr:mathme:v:96:y:2022:i:1:d:10.1007_s00186-022-00776-y
    DOI: 10.1007/s00186-022-00776-y
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    References listed on IDEAS

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