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Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints

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Abstract

The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.

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Acknowledgments

We would like to thank the two anonymous reviewers for their careful reading of this manuscript. Their comments and suggestions certainly improved the readability of this paper.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 , Berlin, Germany

    Michael Hintermüller & Thomas M. Surowiec

  2. Department of Mathematics, Wayne State University, 1150 Faculty/Administration Building, 656 W. Kirby, Detroit, MI, 48202, USA

    Boris S. Mordukhovich

Authors
  1. Michael Hintermüller
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  2. Boris S. Mordukhovich
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  3. Thomas M. Surowiec
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Corresponding author

Correspondence to Michael Hintermüller.

Additional information

The first and third authors would like to acknowledge the financial support from the DFG Research Center MATHEON Project C28, the SPP 1253 “Optimization with Partial Differential Equations”, and the START Project Y 305 “Interfaces and Free Boundaries” funded by the Austrian Ministry of Science and Education and administered by the Austrian Science Fund FWF. The second author would especially like to thank the USA National Science Foundation for their support under grant DMS-1007132, the Australian Research Council under grant DP-12092508, and the Portuguese Foundation of Science and Technologies under grant MAT/11109.

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Hintermüller, M., Mordukhovich, B.S. & Surowiec, T.M. Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Program. 146, 555–582 (2014). https://doi.org/10.1007/s10107-013-0704-6

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  • DOI: https://doi.org/10.1007/s10107-013-0704-6

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