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

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.

Keywords

Cardinality constraints Constraint qualifications  Optimality conditions KKT conditions Strongly stationary points 

Mathematics Subject Classification

90C30 90C11 90B80 90C46 

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bazaraa, M.S., Shetty, C.M.: Foundations of Optimization. Lecture Notes in Economics and Mathematical Systems. Springer (1976)Google Scholar
  3. 3.
    Beck, A., Eldar, Y.C.: Sparsity constrained nonlinear optimization: optimality conditions and algorithms. SIAM J. Optim. 23, 1480–1509 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)MATHGoogle Scholar
  5. 5.
    Bertsekas, D.P., Ozdaglar, A.E.: Pseudonormality and a Lagrange multiplier theory for constrained optimization. J. Optim. Theory Appl. 114, 287–343 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43, 1–2 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bienstock, D.: Computational study of a family of mixed-integer quadratic programming problems. Math. Program. 74, 121–140 (1996)MathSciNetMATHGoogle Scholar
  8. 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. 9.
    Candès, E.J., Wakin, M.B.: An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 21–30 (2008)CrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefMATHGoogle Scholar
  11. 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)Google Scholar
  12. 12.
    Flegel, M.L., Kanzow, C.: Abadie-type constraint qualification for mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 124, 595–614 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Luo, Z.-Q., Pang, J.-S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefMATHGoogle Scholar
  15. 15.
    Meng, K.W., Yang, X.Q.: Optimality conditions via exact penalty functions. SIAM J. Optim. 20, 3208–3231 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Miller, A.: Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, Boca Raton (2002)CrossRefMATHGoogle Scholar
  17. 17.
    Murray, W., Shek, H.: A local relaxation method for the cardinality-constrained portfolio optimization problem. Comput. Optim. Appl. 53, 681–709 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)Google Scholar
  19. 19.
    Outrata, J.V.: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24, 627–644 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Outrata, J.V., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998)CrossRefMATHGoogle Scholar
  21. 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)MathSciNetCrossRefMATHGoogle Scholar
  22. 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)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. A Series of Comprehensive Studies in Mathematics, vol. 317. Springer (1998)Google Scholar
  24. 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)CrossRefGoogle Scholar
  25. 25.
    Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 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)CrossRefMATHGoogle Scholar
  27. 27.
    Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)MathSciNetCrossRefMATHGoogle Scholar
  29. 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)

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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