Stability of generalized equations under nonlinear perturbations

Original Paper
  • 65 Downloads

Abstract

This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.

Keywords

Generalized equation Nonlinear perturbation Local Lipschitz-like property Normal cone mapping Coderivative Partial second-order subdifferential 

Notes

Acknowledgements

This work was done when the authors were working at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank the VIASM for hospitality and kind support.

References

  1. 1.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Dordrecht (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Huy, N.Q., Yao, J.-C.: Exact formulae for coderivatives of normal cone mappings to perturbed polyhedral convex sets. J. Optim. Theory Appl. 157, 25–43 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Lee, G.M., Yen, N.D.: Coderivatives of a Karush–Kuhn–Tucker point set map and applications. Nonlinear Anal. 95, 191–201 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Lee, G.M., Yen, N.D.: Fréchet and normal coderivatives of implicit multifunctions. Appl. Anal. 90, 1011–1027 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley, Chichester (1999)MATHGoogle Scholar
  9. 9.
    Mordukhovich, B.S.: Sensitivity Analysis in Nonsmooth Optimization. Theoretical Aspects of Industrial Design. SIAM, Philadelphia (1992)Google Scholar
  10. 10.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer, Berlin (2006)Google Scholar
  11. 11.
    Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Nam, N.M.: Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2282 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Qui, N.T.: Generalized differentiation of a class of normal cone operators. J. Optim. Theory Appl. 161, 398–429 (2014)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Qui, N.T.: Linearly perturbed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Qui, N.T.: New results on linearly perturbed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Qui, N.T.: Nonlinear perturbations of polyhedral normal cone mappings and affine variational inequalities. J. Optim. Theory Appl. 153, 98–122 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Qui, N.T.: Stability for trust-region methods via generalized differentiation. J. Glob. Optim. 59, 139–164 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Qui, N.T.: Upper and lower estimates for a Fréchet normal cone. Acta Math. Vietnam 36, 601–610 (2011)MathSciNetMATHGoogle Scholar
  19. 19.
    Qui, N.T.: Variational inequalities over Euclidean balls. Math. Methods Oper. Res. 78, 243–258 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Qui, N.T., Yen, N.D.: A class of linear generalized equations. SIAM J. Optim. 24, 210–231 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Robinson, S.M.: Generalized equations and their solutions. I. Basic theory. Math. Program. Stud. 10, 128–141 (1979)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  23. 23.
    Trang, N.T.Q.: Lipschitzian stability of parametric variational inequalities over perturbed polyhedral convex sets. Optim. Lett. 6, 749–762 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality. I. Basic calculations. Acta Math. Vietnam 34, 157–172 (2009)MathSciNetMATHGoogle Scholar
  25. 25.
    Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality. II. Applications. Pac. J. Optim. 5, 493–506 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Information and Communication TechnologyCan Tho UniversityCan ThoVietnam
  2. 2.Department of MathematicsHanoi Pedagogical University 2Phuc YenVietnam

Personalised recommendations