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On Linear and Linearized Generalized Semi-Infinite Optimization Problems

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  • 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
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    Citations

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    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. 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. 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.
    5. 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.
    6. 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.
    7. 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.
    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.

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