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A transformation-based discretization method for solving general semi-infinite optimization problems

Author

Listed:
  • Jan Schwientek

    (Fraunhofer Institute for Industrial Mathematics (ITWM))

  • Tobias Seidel

    (Fraunhofer Institute for Industrial Mathematics (ITWM))

  • Karl-Heinz Küfer

    (Fraunhofer Institute for Industrial Mathematics (ITWM))

Abstract

Discretization methods are commonly used for solving standard semi-infinite optimization (SIP) problems. The transfer of these methods to the case of general semi-infinite optimization (GSIP) problems is difficult due to the $$\mathbf {x}$$ x -dependence of the infinite index set. On the other hand, under suitable conditions, a GSIP problem can be transformed into a SIP problem. In this paper we assume that such a transformation exists globally. However, this approach may destroy convexity in the lower level, which is very important for numerical methods. We present in this paper a solution approach for GSIP problems, which cleverly combines the above mentioned two techniques. It is shown that the convergence results for discretization methods in the case of SIP problems can be transferred to our transformation-based discretization method under suitable assumptions on the transformation. Finally, we illustrate the operation of our approach as well as its performance on several examples, including a problem of volume-maximal inscription of multiple variable bodies into a larger fixed body, which has never before been considered as a GSIP test problem.

Suggested Citation

  • Jan Schwientek & Tobias Seidel & Karl-Heinz Küfer, 2021. "A transformation-based discretization method for solving general semi-infinite optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 83-114, February.
    • Handle: RePEc:spr:mathme:v:93:y:2021:i:1:d:10.1007_s00186-020-00724-8
    DOI: 10.1007/s00186-020-00724-8
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    References listed on IDEAS

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    1. Still, G., 1999. "Generalized semi-infinite programming: Theory and methods," European Journal of Operational Research, Elsevier, vol. 119(2), pages 301-313, December.
    2. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    3. Peter Kirst & Oliver Stein, 2019. "Global optimization of generalized semi-infinite programs using disjunctive programming," Journal of Global Optimization, Springer, vol. 73(1), pages 1-25, January.
    4. Alexander Mitsos & Angelos Tsoukalas, 2015. "Global optimization of generalized semi-infinite programs via restriction of the right hand side," Journal of Global Optimization, Springer, vol. 61(1), pages 1-17, January.
    5. Hatim Djelassi & Moll Glass & Alexander Mitsos, 2019. "Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints," Journal of Global Optimization, Springer, vol. 75(2), pages 341-392, October.
    6. Winterfeld, Anton, 2008. "Application of general semi-infinite programming to lapidary cutting problems," European Journal of Operational Research, Elsevier, vol. 191(3), pages 838-854, December.
    7. Oliver Stein, 2006. "A semi-infinite approach to design centering," Springer Optimization and Its Applications, in: Stephan Dempe & Vyacheslav Kalashnikov (ed.), Optimization with Multivalued Mappings, pages 209-228, Springer.
    8. Stuart M. Harwood & Paul I. Barton, 2017. "How to solve a design centering problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 215-254, August.
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