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Some Remarks on Stability of Generalized Equations

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Abstract

The paper concerns the computation of the graphical derivative and the regular (Fréchet) coderivative of the solution map to a class of generalized equations, where the multivalued term amounts to the regular normal cone to a (possibly nonconvex) set given by C 2 inequalities. Instead of the linear independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian–Fromovitz and the constant rank qualification conditions. Based on the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constraints are derived, and a workable characterization of the isolated calmness of the considered solution map is provided.

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Acknowledgements

The research was partially supported by the Grant Agency of the Czech Academy of Sciences, project IAA 100750802; the Grant Agency of the Czech Republic, project P201/12/0671; the DFG Research Center Matheon “Mathematics for key technologies” in Berlin; and the Australian Research Council, project DP110102011.

The authors wish to thank the anonymous referee for the careful reading of the paper and valuable comments and suggestions.

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Authors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, 10117, Berlin, Germany

    René Henrion

  2. Centre for Informatics and Applied Optimization, School of Science, Information Technology and Engineering, University of Ballarat, POB 663, Ballarat, Vic, 3350, Australia

    Alexander Y. Kruger

  3. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, 18208, Prague, Czech Republic

    Jiří V. Outrata

Authors
  1. René Henrion
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  2. Alexander Y. Kruger
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  3. Jiří V. Outrata
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Correspondence to Alexander Y. Kruger.

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Henrion, R., Kruger, A.Y. & Outrata, J.V. Some Remarks on Stability of Generalized Equations. J Optim Theory Appl 159, 681–697 (2013). https://doi.org/10.1007/s10957-012-0147-x

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  • DOI: https://doi.org/10.1007/s10957-012-0147-x

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