IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v101y2001i1p191-20810.1023-a1010972524021.html
   My bibliography  Save this article

On Linear and Linearized Generalized Semi-Infinite Optimization Problems

Author

Listed:
  • Jan-J. Rückmann
  • Oliver Stein

Abstract

We consider the local and global topological structure of the feasible set M of a generalized semi-infinite optimization problem. Under the assumption that the defining functions for M are affine-linear with respect to the index variable and separable with respect to the index and the state variable, M can globally be written as the finite union of certain open and closed sets. Here, it is not necessary to impose any kind of constraint qualification on the lower level problem. In fact, these sets are level sets of the lower level Lagrangian, and the open sets are generated exactly by Lagrange multiplier vectors with vanishing entry corresponding to the lower level objective function. This result gives rise to a first order necessary optimality condition for the considered generalized semi-infinite problem. Finally it is shown that the description of M by open and closed level sets of the lower level Lagrangian locally carries over to points of the so-called mai-type, where neither the linearity nor the separability assumption is satisfied. Copyright Kluwer Academic Publishers 2001

Suggested Citation

  • Jan-J. Rückmann & Oliver Stein, 2001. "On Linear and Linearized Generalized Semi-Infinite Optimization Problems," Annals of Operations Research, Springer, vol. 101(1), pages 191-208, January.
    • Handle: RePEc:spr:annopr:v:101:y:2001:i:1:p:191-208:10.1023/a:1010972524021
    DOI: 10.1023/A:1010972524021
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1023/A:1010972524021
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1023/A:1010972524021?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. Geletu, Abebe & Hoffmann, Armin, 2004. "A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization," European Journal of Operational Research, Elsevier, vol. 157(1), pages 3-15, August.
    4. 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.
    5. 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.
    6. Le Thanh Tung, 2022. "Karush–Kuhn–Tucker optimality conditions and duality for multiobjective semi-infinite programming with vanishing constraints," Annals of Operations Research, Springer, vol. 311(2), pages 1307-1334, April.
    7. 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.
    8. Oliver Stein, 2012. "Comments on: Stability in linear optimization and related topics. A personal tour," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 265-266, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:annopr:v:101:y:2001:i:1:p:191-208:10.1023/a:1010972524021. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.