Skip to main content
SpringerLink
Log in

Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

By applying some theorems of Levy and Mordukhovich (Math Program 99:311–327, 2004) and other related results, we estimate 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 the obtained formulas, we derive necessary and sufficient conditions for the local Lipschitz-like property of the stationary point set map. This leads us to new insights into the preceding deep investigations of Levy and Mordukhovich in the above-cited paper and of Qui (J Optim Theory Appl 161:398–429, 2014, J Glob Optim 65:615–635, 2016).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Levy, A.B., Mordukhovich, B.S.: Coderivatives in parametric optimization. Math. Program. 99, 311–327 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Yen, N.D., Yao, J.-C.: Point-based sufficient conditions for metric regularity of implicit multifunctions. Nonlinear Anal. 70, 2806–2815 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borwein, J.M., Zhuang, D.M.: Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps. J. Math. Anal. Appl. 134, 441–459 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Penot, J.-P.: Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13, 629–643 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–36 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Célia, J.-A., Alain, P.: A variant of Newton’s method for generalized equations. Rev. Colombiana Mat. 39, 97–112 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Dontchev, A.L.: Uniform convergence of the Newton method for Aubin continuous maps. Well-posedness and stability of variational problems. Serdica Math. J. 22, 283–296 (1996)

    MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Robinson, S.M.: Stability theory for systems of inequalities, II. Differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, G.M., Yen, N.D.: Coderivatives of a Karush-Kuhn-Tucker point set map and applications. Nonlinear Anal. 95, 191–201 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Mordukhovich, B.S., Nghia, T.T.A.: Full Lipschitzian and Hölderian stability in optimization with applications to mathematical programming and optimal control. SIAM J. Optim. 24, 1344–1381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mordukhovich, B.S., Nghia, T.T.A.: Local monotonicity and full stability for parametric variational systems. SIAM J. Optim. 26, 1032–1059 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mordukhovich, B.S., Nghia, T.T.A., Rockafellar, R.T.: Full stability in finite-dimensional optimization: Mordukhovich. Math. Oper. Res. 40, 226–252 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ban, L., Song, W.: Linearly perturbed generalized polyhedral normal cone mappings and applications. Optimization 65, 9–34 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-oder analysis of polyhedral systems in finite and infinite dimensions and applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lu, S., Robinson, S.M.: Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. 33, 689–711 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nam, N.M.: Coderivatives of normal cone mappings and Lipschitzian stability of parametric variational inequalities. Nonlinear Anal. 73, 2271–2282 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qui, N.T.: Upper and lower estimates for a Fréchet normal cone. Acta Math. Vietnam 36, 601–610 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Qui, N.T.: Linearly perturbed polyhedral normal cone mappings and applications. Nonlinear Anal. 74, 1676–1689 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Qui, N.T.: New results on linearly perturbed polyhedral normal cone mappings. J. Math. Anal. Appl. 381, 352–364 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Robinson, S.M.: Solution continuity in monotone affine variational inequalities. SIAM J. Optim. 18, 1046–1060 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Trang, N.T.Q.: A note on Lipschitzian stability of variational inequalities over perturbed polyhedral convex sets. Optim. Lett. 10, 1221–1231 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    MathSciNet  MATH  Google Scholar 

  33. 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)

    MathSciNet  MATH  Google Scholar 

  34. Qui, N.T.: Nonlinear perturbations of polyhedral normal cone mappings and affine variational inequalities. J. Optim. Theory Appl. 153, 98–122 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Guo, L., Lin, G.-H., Ye, J.J.: Stability analysis for parametric mathematical programs with geometric constraints and its applications. SIAM J. Optim. 22, 1151–1176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  37. Huyen, D.T.K., Yen, N.D.: Coderivatives and the solution map of a linear constraint system. SIAM J. Optim. 26, 986–1007 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill Book Co., New York (1976)

    MATH  Google Scholar 

Download references

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. Examples 3.4 and 4.2 in this paper present our solutions to two open questions raised by one of the referees. In addition, Remarks 3.2 and 3.3 are based on some comments of that referee.

Author information

Authors and Affiliations

  1. Graduate Training Center, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Duong Thi Kim Huyen

  2. Center for General Education, China Medical University, Taichung, Taiwan

    Jen-Chih Yao

  3. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Nguyen Dong Yen

Authors
  1. Duong Thi Kim Huyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jen-Chih Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nguyen Dong Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Dong Yen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huyen, D.T.K., Yao, JC. & Yen, N.D. Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability. J Optim Theory Appl 180, 91–116 (2019). https://doi.org/10.1007/s10957-018-1294-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1294-5

Keywords

Mathematics Subject Classification

Search

Navigation