Skip to main content
SpringerLink
Log in

Constraint qualifications and optimality conditions for optimization problems with cardinality constraints

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Andreani, R., Martínez, J.M., Schuverdt, M.L.: The CPLD condition of Qi and Wei implies the quasinormality qualification. J. Optim. Theory Appl. 125, 473–485 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bazaraa, M.S., Shetty, C.M.: Foundations of Optimization. Lecture Notes in Economics and Mathematical Systems. Springer (1976)

  3. Beck, A., Eldar, Y.C.: Sparsity constrained nonlinear optimization: optimality conditions and algorithms. SIAM J. Optim. 23, 1480–1509 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  5. Bertsekas, D.P., Ozdaglar, A.E.: Pseudonormality and a Lagrange multiplier theory for constrained optimization. J. Optim. Theory Appl. 114, 287–343 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43, 1–2 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)

    MathSciNet  MATH  Google Scholar 

  8. Burdakov, O.P., Kanzow, C., Schwartz, A.: Mathematical programs with cardinality constraints: reformulation by complementarity-type conditions and a regularization method. SIAM J. Optim. to appear (doi:10.1137/140978077)

  9. Candès, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008)

    Article  Google Scholar 

  10. Di Lorenzo, D., Liuzzi, G., Rinaldi, F., Schoen, F., Sciandrone, M.: A concave optimization-based approach for sparse portfolio selection. Optim. Methods Softw. 27, 983–1000 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feng, M., Mitchell, J.E., Pang, J.-S., Shen, X., Wächter, A.: Complementarity formulation of \( \ell _0 \)-norm optimization problems. Industrial Engineering and Management Sciences. Technical Report. Northwestern University, Evanston, IL, USA (2013)

  12. Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124, 595–614 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flegel, M.L., Kanzow, C., Outrata, J.V.: Optimality conditions for disjunctive programs with application to mathematical programs with equilibrium constraints. Set-Valued Anal. 15, 139–162 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  15. Meng, K.W., Yang, X.Q.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20, 3208–3231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miller, A.: Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, Boca Raton (2002)

    Book  MATH  Google Scholar 

  17. Murray, W., Shek, H.: A local relaxation method for the cardinality-constrained portfolio optimization problem. Comput. Optim. Appl. 53, 681–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)

  19. Outrata, J.V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627–644 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  21. Pang, J.-S., Fukushima, M.: Complementarity constraint qualifications and simplified B-stationarity conditions for mathematical programs with equilibrium constraints. Comput. Optim. Appl. 13, 111–136 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qi, L., Wei, Z.: On the constant positive linear dependence constraint qualification and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. A Series of Comprehensive Studies in Mathematics, vol. 317. Springer (1998)

  24. Ruiz-Torrubiano, R., García-Moratilla, S., Suárez, A.: Optimization problems with cardinality constraints. In: Tenne, Y., Goh, C.-K. (eds.) Computational Intelligence in Optimization, pp. 105–130. Springer, Berlin (2010)

    Chapter  Google Scholar 

  25. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sun, X., Zheng, X., Li, D.: Recent advances in mathematical programming with semi-continuous variables and cardinality constraint. J. Oper. Res. Soc. China 1, 55–77 (2013)

    Article  MATH  Google Scholar 

  27. Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zheng, X., Sun, X., Li, D., Sun, J.: Successive convex approximations to cardinality-constrained convex programs: a piecewise-linear DC approach. Comput. Optim. Appl. to appear (doi:10.1007/s10589-013-9582-3)

Download references

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.

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Kanzow
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexandra Schwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

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.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

Mathematics Subject Classification

Search

Navigation