Notes on Some Constraint Qualifications for Mathematical Programs with Equilibrium Constraints

Article

Abstract

We study the constraint qualifications for mathematical programs with equilibrium constraints (MPEC). Firstly, we investigate the weakest constraint qualifications for the Bouligand and Mordukhovich stationarities for MPEC. Then, we show that the MPEC relaxed constant positive linear dependence condition can ensure any locally optimal solution to be Mordukhovich stationary. Finally, we give the relations among the existing MPEC constraint qualifications.

Keywords

Mathematical program with equilibrium constraints Constraint qualification Bouligand stationarity Mordukhovich stationarity 

References

  1. 1.
    Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) CrossRefGoogle Scholar
  3. 3.
    Outrata, J.V., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic Publishers, Boston (1998) MATHGoogle Scholar
  4. 4.
    Fukushima, M., Lin, G.H.: Smoothing methods for mathematical programs with equilibrium constraints. In: Proceedings of the ICKS’04, pp. 206–213. IEEE Comput. Soc., Los Alamitos (2004) Google Scholar
  5. 5.
    Flegel, M.L., Kanzow, C.: A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, vol. 2, pp. 111–122. Springer, New York (2006) CrossRefGoogle Scholar
  6. 6.
    Kanzow, C., Schwartz, A.: Mathematical programs with equilibrium constraints: enhanced fritz john conditions, new constraint qualifications and improved exact penalty results. SIAM J. Optim. 20, 2730–2753 (2010) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Scheel, H.S., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25, 1–22 (2000) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Ye, J.J.: Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. 10, 943–962 (2000) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17, 259–286 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lin, G.H., Guo, L., Ye, J.J.: Solving mathematical programs with equilibrium constraints as constrained equations (submitted) Google Scholar
  13. 13.
    Hu, X., Ralph, D.: Convergence of a penalty method for mathematical programming with equilibrium constraints. J. Optim. Theory Appl. 123, 365–390 (2004) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11, 918–936 (2001) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Guo, L., Lin, G.H., Ye, J.J.: Second order optimality conditions for mathematical programs with equilibrium constraint (submitted) Google Scholar
  16. 16.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006) Google Scholar
  17. 17.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998) MATHCrossRefGoogle Scholar
  18. 18.
    Flegel, M.L., Kanzow, C.: On M-stationary points for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 310, 286–302 (2005) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16, 199–227 (2008) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Andreani, R., Haeser, G., Schuverdt, M.L., Siliva, J.S.: A relaxed constant positive linear dependence constraint qualification and applications. Math. Program. (2011). doi:10.1007/s10107-011-0456-0 Google Scholar
  21. 21.
    Ye, J.J., Zhang, J.: Enhanced Karush–Kuhn–Tucker condition for mathematical programs with equilibrium constraints (submmited) Google Scholar
  22. 22.
    Flegel, M.L., Kanzow, C.: On the guignard constraint qualification for mathematical programs with equilibrium constraints. Optimization 54, 517–534 (2005) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 2. Necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. (2011). doi:10.1007/s10107-011-0488-5 MATHGoogle Scholar
  25. 25.
    Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties (submitted) Google Scholar
  26. 26.
    Minchenko, L., Stakhovski, S.: Parametric nonlinear programming problems under the relaxed constant rank condition. SIAM J. Optim. 21, 314–332 (2011) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical SciencesDalian University of TechnologyDalianChina

Personalised recommendations