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On Optimality Conditions for Generalized Semi-Infinite Programming Problems

Author

Listed:
  • G. Stein

    (Aachen University of Technology)

  • G. Still

    (University of Twente)

Abstract

Generalized semi-infinite optimization problems (GSIP) are considered. We generalize the well-known optimality conditions for minimizers of order one in standard semi-infinite programming to the GSIP case. We give necessary and sufficient conditions for local minimizers of order one without the assumption of local reduction. The necessary conditions are derived along the same lines as the first-order necessary conditions for GSIP in a recent paper of Jongen, Rückmann, and Stein (Ref. 1) by assuming the so-called extended Mangasarian–Fromovitz constraint qualification. Using the ideas of a recent paper of Rückmann and Shapiro, we give short proofs of necessary and sufficient optimality conditions for minimizers of order one under the additional assumption of the Mangasarian–Fromovitz constraint qualification at all local minimizers of the so-called lower-level problem.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:104:y:2000:i:2:d:10.1023_a:1004622015901
    DOI: 10.1023/A:1004622015901
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    References listed on IDEAS

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    1. 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.
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    Citations

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    Cited by:

    1. O. Stein, 2004. "On Constraint Qualifications in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 647-671, June.
    2. Stein, Oliver & Still, Georg, 2002. "On generalized semi-infinite optimization and bilevel optimization," European Journal of Operational Research, Elsevier, vol. 142(3), pages 444-462, November.
    3. Oliver Stein, 2001. "First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Optimization," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 565-582, August.
    4. Kanzi, N. & Nobakhtian, S., 2010. "Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems," European Journal of Operational Research, Elsevier, vol. 205(2), pages 253-261, September.
    5. 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.
    6. Jan Rückmann & Juan Alfredo Gómez, 2006. "On generalized semi-infinite programming," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 1-32, June.
    7. 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.
    8. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    9. Lopez, Marco & Still, Georg, 2007. "Semi-infinite programming," European Journal of Operational Research, Elsevier, vol. 180(2), pages 491-518, July.
    10. 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.
    11. J. J. Ye & S. Y. Wu, 2008. "First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 419-434, May.
    12. T. Q. Son & J. J. Strodiot & V. H. Nguyen, 2009. "ε-Optimality and ε-Lagrangian Duality for a Nonconvex Programming Problem with an Infinite Number of Constraints," Journal of Optimization Theory and Applications, Springer, vol. 141(2), pages 389-409, May.

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