Journal of Global Optimization

, Volume 65, Issue 3, pp 615–635 | Cite as

Coderivatives of implicit multifunctions and stability of variational systems



We establish formulas for computing/estimating the regular and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smooth-boundary constraint sets.


Coderivative Generalized equation Implicit multifunction Local Lipschitz-like property Normal cone operator Variational system 

Mathematics Subject Classification

49J53 49J52 49J40 



A part of this work was done when the author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). The author would like to thank the VIASM for hospitality and kind support. The author is indebted to the handling Editors and the anonymous referees for their valuable remarks and detailed suggestions that have greatly improved the original version of the paper.


  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)MathSciNetMATHGoogle Scholar
  2. 2.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Dordrecht (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Fabian, M., Mordukhovich, B.S.: Sequential normal compactness versus topological normal compactness in variational analysis. Nonlinear Anal. 54, 1057–1067 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Lang, S.: Real and Functional Analysis. Springer, New York (1993)CrossRefMATHGoogle Scholar
  6. 6.
    Le Thi, H.A., Pham Dinh, T.: Solving a class of linearly constrained indefinite quadratic problems by d.c. algorithms. J. Global Optim. 11, 253–285 (1997)Google Scholar
  7. 7.
    Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Behavior of DCA sequences for solving the trust-region subproblem. J. Global Optim. 53, 317–329 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Le Thi, H.A., Pham Dinh, T., Yen, N.D.: Properties of two DC algorithms in quadratic programming. J. Global Optim. 49, 481–495 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ledyaev, Y.S., Zhu, Q.J.: Implicit multifunctions theorems. Set-Valued Anal. 7, 209–238 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lee, G.M., Yen, N.D.: Fréchet and normal coderivatives of implicit multifunctions. Appl. Anal. 90, 1011–1027 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. Ser. A. 99, 311–327 (2004)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mordukhovich, B.S.: Coderivative analysis of variational systems. J. Global Optim. 28, 347–362 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Springer, Berlin (2006)Google Scholar
  15. 15.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol II: Applications. Springer, Berlin (2006)Google Scholar
  16. 16.
    Nam, N.M.: Coderivatives of normal mappings and the Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2282 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pham Dinh, T., Le Thi, H.A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8, 476–505 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Qui, N.T.: Generalized differentiation of a class of normal cone operators. J. Optim. Theory Appl. 161, 398–429 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Qui, N.T.: Linearly perturbed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Qui, N.T.: New results on linearly perturbed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Qui, N.T.: Nonlinear perturbations of polyhedral normal cone mappings and affine variational inequalities. J. Optim. Theory Appl. 153, 98–122 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Qui, N.T.: Stability for trust-region methods via generalized differentiation. J. Global Optim. 59, 139–164 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Qui, N.T.: Variational inequalities over Euclidean balls. Math. Methods Oper. Res. 78, 243–258 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Qui, N.T., Yen, N.D.: A class of linear generalized equations. SIAM J. Optim. 24, 210–231 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Robinson, S.M.: Generalized equations and their solutions. I. Basic theory. Math. Program. Stud. 10, 128–141 (1979)Google Scholar
  26. 26.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  27. 27.
    Rudin, W.: Principles of Mathematical Analysis. McGraw-Hill, New Delhi (1976)MATHGoogle Scholar
  28. 28.
    Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, part 1: basic calculations. Acta Math. Vietnam. 34, 157–172 (2009)MathSciNetMATHGoogle Scholar
  29. 29.
    Yao, J.-C., Yen, N.D.: Coderivative calculation related to a parametric affine variational inequality, part 2: applications. Pacific J. Optim. 5, 493–506 (2009)MathSciNetMATHGoogle Scholar
  30. 30.
    Yao, J.-C., Yen, N.D.: Parametric variational system with a smooth-boundary constraint set. In: Mordukhovich, B.S., Burachik, R.S., Yao, J.-C. (eds.) Variational Analysis and Generalized Differentiation in Optimization and Control, vol. 47, pp. 205–221. Springer, New York (2010). Series “Optimization and Its Applications”Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.College of Information and Communication TechnologyCan Tho UniversityCan ThoVietnam

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