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Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets

Author

Listed:
  • Olga Kostyukova

    (National Academy of Sciences of Belarus)

  • Tatiana Tchemisova

    (University of Aveiro)

Abstract

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems.

Suggested Citation

  • Olga Kostyukova & Tatiana Tchemisova, 2017. "Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 76-103, October.
    • Handle: RePEc:spr:joptap:v:175:y:2017:i:1:d:10.1007_s10957-017-1150-z
    DOI: 10.1007/s10957-017-1150-z
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    References listed on IDEAS

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    1. Sanjay Mehrotra & David Papp, 2013. "A cutting surface algorithm for semi-infinite convex programming with an application to moment robust optimization," Papers 1306.3437, arXiv.org, revised Aug 2014.
    2. Xiaoqi Yang & Zhangyou Chen & Jinchuan Zhou, 2016. "Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 984-1012, June.
    3. O. I. Kostyukova & T. V. Tchemisova & S. A. Yermalinskaya, 2010. "Convex Semi-Infinite Programming: Implicit Optimality Criterion Based on the Concept of Immobile Indices," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 325-342, May.
    4. G. Stein & G. Still, 2000. "On Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 104(2), pages 443-458, February.
    5. J. J. Rückmann & A. Shapiro, 1999. "First-Order Optimality Conditions in Generalized Semi-Infinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 677-691, June.
    6. Peter Kirst & Oliver Stein, 2016. "Solving Disjunctive Optimization Problems by Generalized Semi-infinite Optimization Techniques," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1079-1109, June.
    7. P. Parpas & B. Rustem, 2009. "An Algorithm for the Global Optimization of a Class of Continuous Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 461-473, May.
    8. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    9. O. Kostyukova & T. Tchemisova, 2012. "Implicit optimality criterion for convex SIP problem with box constrained index set," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 475-502, July.
    10. Faizan Ahmed & Mirjam Dür & Georg Still, 2013. "Copositive Programming via Semi-Infinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 322-340, November.
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    Cited by:

    1. O. I. Kostyukova & T. V. Tchemisova, 2022. "On strong duality in linear copositive programming," Journal of Global Optimization, Springer, vol. 83(3), pages 457-480, July.
    2. Olga Kostyukova & Tatiana Tchemisova, 2021. "Structural Properties of Faces of the Cone of Copositive Matrices," Mathematics, MDPI, vol. 9(21), pages 1-21, October.

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