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Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 117–139 | Cite as

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 2: Robinson Stability

  • Duong Thi Kim Huyen
  • Jen-Chih Yao
  • Nguyen Dong YenEmail author
Article

Abstract

In Part 1 of this paper, we have estimated the Fréchet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.

Keywords

Smooth parametric optimization problem Smooth functional constraint Stationary point set map Robinson stability Coderivative 

Mathematics Subject Classification

49K40 49J53 90C31 90C20 

Notes

Acknowledgements

This work was supported by National Foundation for Science & Technology Development (Vietnam) and the Grant MOST 105-2115-M-039-002-MY3 (Taiwan). The authors are grateful to the anonymous referees for their careful readings, encouragement, and valuable suggestions. Section 5 is based on the comments made by one of the referees.

References

  1. 1.
    Huyen, D.T.K., Yao, J.-C., Yen, N.D.: Sensitivity analysis of a stationary point set map under total perturbations. Part 1: Lipschitzian stability. J. Optim. Theory Appl.  https://doi.org/10.1007/s10957-018-1294-5
  2. 2.
    Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. 99, 311–327 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Qui, N.T.: Generalized differentiation of a class of normal cone operators. J. Optim. Theory Appl. 161, 398–429 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Qui, N.T.: Coderivatives of implicit multifunctions and stability of variational systems. J. Glob. Optim. 65, 615–635 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  7. 7.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–36 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Huyen, D.T.K., Yen, N.D.: Coderivatives and the solution map of a linear constraint system. SIAM J. Optim. 26, 986–1007 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. SIAM J. Optim. 27, 438–465 (2017)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lee, G.M., Yen, N.D.: Fréchet and normal coderivatives of implicit multifunctions. Appl. Anal. 90, 1011–1027 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yen, N.D., Yao, J.-C.: Point-based sufficient conditions for metric regularity of implicit multifunctions. Nonlinear Anal. 70, 2806–2815 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, G.M., Tam, N.N., Yen, N.D.: Stability of linear-quadratic minimization over Euclidean balls. SIAM J. Optim. 22, 936–952 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, G.M., Yen, N.D.: Coderivatives of a Karush–Kuhn–Tucker point set map and applications. Nonlinear Anal. 95, 191–201 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Qui, N.T., Yen, N.D.: A class of linear generalized equations. SIAM J. Optim. 24, 210–231 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dontchev, A., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Duong Thi Kim Huyen
    • 1
  • Jen-Chih Yao
    • 2
  • Nguyen Dong Yen
    • 3
    Email author
  1. 1.Graduate Training Center, Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.Center for General EducationChina Medical UniversityTaichungTaiwan
  3. 3.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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