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Mathematical Programs with Complementarity Constraints in Banach Spaces

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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.

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Acknowledgments

The author would like to thank Radu Ioan Boţ for the idea leading to the counterexample at the end of Sect. 4.

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Authors and Affiliations

  1. Faculty of Mathematics, Professorship Numerical Mathematics (Partial Differential Equations), Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Gerd Wachsmuth

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  1. Gerd Wachsmuth
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Correspondence to Gerd Wachsmuth.

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Communicated by Masao Fukushima.

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Wachsmuth, G. Mathematical Programs with Complementarity Constraints in Banach Spaces. J Optim Theory Appl 166, 480–507 (2015). https://doi.org/10.1007/s10957-014-0695-3

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