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Mathematical Programs with Complementarity Constraints in Banach Spaces

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  • Gerd Wachsmuth

    (Technische Universität Chemnitz)

Abstract

We consider optimization problems in Banach spaces involving a complementarity constraint, defined by a convex cone K. By transferring the local decomposition approach, we define strong stationarity conditions and provide a constraint qualification, under which these conditions are necessary for optimality. To apply this technique, we provide a new uniqueness result for Lagrange multipliers in Banach spaces. In the case that the cone K is polyhedral, we show that our strong stationarity conditions possess a reasonable strength. Finally, we generalize to the case where K is not a cone and apply the theory to two examples.

Suggested Citation

  • Gerd Wachsmuth, 2015. "Mathematical Programs with Complementarity Constraints in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 480-507, August.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:2:d:10.1007_s10957-014-0695-3
    DOI: 10.1007/s10957-014-0695-3
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    References listed on IDEAS

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    1. M.L. Flegel & C. Kanzow, 2005. "Abadie-Type Constraint Qualification for Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 124(3), pages 595-614, March.
    2. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, February.
    3. Jean-Baptiste Hiriart-Urruty & Jérôme Malick, 2012. "A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 551-577, June.
    4. K. Krumbiegel & A. Rösch, 2009. "A virtual control concept for state constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 43(2), pages 213-233, June.
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