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Constraint qualifications and optimality conditions for optimization problems with cardinality constraints

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Abstract

This paper considers optimization problems with cardinality constraints. Based on a recently introduced reformulation of this problem as a nonlinear program with continuous variables, we first define some problem-tailored constraint qualifications and then show how these constraint qualifications can be used to obtain suitable optimality conditions for cardinality constrained problems. Here, the (KKT-like) optimality conditions hold under much weaker assumptions than the corresponding result that is known for the somewhat related class of mathematical programs with complementarity constraints.

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Acknowledgments

The authors would like to thank both referees for their very detailed comments which helped quite a bit to improve the presentation of the paper.

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Authors and Affiliations

  1. Institute of Information Theory and Automation, Czech Academy of Sciences, Pod Vodárenskou věží 4, 182 08, Prague, Czech Republic

    Michal Červinka

  2. Faculty of Social Sciences, Institute of Economic Studies, Charles University in Prague, Opletalova 26, 110 00, Prague, Czech Republic

    Michal Červinka

  3. Institute of Mathematics, University of Würzburg, Emil-Fischer-Straße 30, 97074, Würzburg, Germany

    Christian Kanzow

  4. Graduate School of Computational Engineering, TU Darmstadt, Dolivostraße 15, 64293, Darmstadt, Germany

    Alexandra Schwartz

Authors
  1. Michal Červinka
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  2. Christian Kanzow
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  3. Alexandra Schwartz
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Corresponding author

Correspondence to Alexandra Schwartz.

Additional information

This research was partially supported by Grants P402/12/1309 and 15-00735S of the Grant Agency of the Czech Republic and by the Graduate School of Computational Engineering at TU Darmstadt.

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Červinka, M., Kanzow, C. & Schwartz, A. Constraint qualifications and optimality conditions for optimization problems with cardinality constraints. Math. Program. 160, 353–377 (2016). https://doi.org/10.1007/s10107-016-0986-6

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  • DOI: https://doi.org/10.1007/s10107-016-0986-6

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