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

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.

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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)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Qui, N.T.: Generalized differentiation of a class of normal cone operators. J. Optim. Theory Appl. 161, 398–429 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Qui, N.T.: Coderivatives of implicit multifunctions and stability of variational systems. J. Glob. Optim. 65, 615–635 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)

    Google Scholar 

  7. 7.

    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. SIAM J. Optim. 27, 438–465 (2017)

    MathSciNet  Article  MATH  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)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Lee, G.M., Yen, N.D.: Fréchet and normal coderivatives of implicit multifunctions. Appl. Anal. 90, 1011–1027 (2011)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Qui, N.T., Yen, N.D.: A class of linear generalized equations. SIAM J. Optim. 24, 210–231 (2014)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Dontchev, A., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis. Springer, Berlin (2009)

    Google Scholar 

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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.

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Correspondence to Nguyen Dong Yen.

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Huyen, D.T.K., Yao, J. & Yen, N.D. Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 2: Robinson Stability. J Optim Theory Appl 180, 117–139 (2019). https://doi.org/10.1007/s10957-018-1295-4

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Keywords

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

Mathematics Subject Classification

  • 49K40
  • 49J53
  • 90C31
  • 90C20