Mathematical Programming

, Volume 160, Issue 1–2, pp 353–377 | Cite as

Constraint qualifications and optimality conditions for optimization problems with cardinality constraints

  • Michal Červinka
  • Christian Kanzow
  • Alexandra Schwartz
Full Length Paper Series A


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.


Cardinality constraints Constraint qualifications  Optimality conditions KKT conditions Strongly stationary points 

Mathematics Subject Classification

90C30 90C11 90B80 90C46 



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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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Institute of Information Theory and AutomationCzech Academy of SciencesPragueCzech Republic
  2. 2.Faculty of Social Sciences, Institute of Economic StudiesCharles University in PraguePragueCzech Republic
  3. 3.Institute of MathematicsUniversity of WürzburgWürzburgGermany
  4. 4.Graduate School of Computational EngineeringTU DarmstadtDarmstadtGermany

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