Skip to main content

The Stationary Point Set Map in General Parametric Optimization Problems

Set-Valued and Variational Analysis volume 30, pages 305–327 (2022)Cite this article

Abstract

The present paper shows how the linear independence constraint qualification (LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a general C2-smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91–116 (Part 1); 117–139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953–986; Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Alt, W.: Stability of solutions for a class of nonlinear cone constrained optimization problems. I. Basic Theory. Numer. Funct. Anal. Optim. 10, 1053–1064 (1989)

    MathSciNet  Article  Google Scholar 

  2. Alt, W.: Local stability of solutions to differentiable optimization problems in Banach spaces. J. Optim. Theory Appl. 70, 443–466 (1991)

    MathSciNet  Article  Google Scholar 

  3. Aubin, J.-P., Frankowska, H.: Set-valued Analysis. Birkhäuser Boston (1990)

  4. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  Google Scholar 

  5. Chieu, N.H., Yao, J.-C., Yen, N.D.: Relationships between Robinson robust stability and Lipschitz-like behavior of implicit multifunctions. Nonlinear Anal. 72, 3594–3601 (2010)

    MathSciNet  Article  Google Scholar 

  6. Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust-Region Methods. Society for Industrial and Applied Mathematics (SIAM) and Mathematical Programming Society (MPS), Philadelphia (2000)

  7. Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. Math. Programming Stud. 19, 101–119 (1982)

    MathSciNet  Article  Google Scholar 

  8. Gauvin, J., Dubeau, F.: Some examples and counterexamples for the stability analysis of nonlinear programming problems. Math. Programming Stud. 21, 69–78 (1983)

    MathSciNet  Article  Google Scholar 

  9. Gauvin, J., Tolle, W.J.: Differential stability in nonlinear programming. SIAM J. Control Optimization. 15, 294–311 (1977)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  11. 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  Google Scholar 

  12. Huyen, D.T.K., Yao, J.-C., Yen, N.D.: Sensitivity analysis of an optimization problem under total perturbations. Part 1: Lipschitzian stability. J. Optim. Theory Appl. 180, 91–116 (2019)

    MathSciNet  Article  Google Scholar 

  13. Huyen, D.T.K., Yao, J.-C., Yen, N.D.: Sensitivity analysis of an optimization problem under total perturbations. Part 2: Robinson stability. J. Optim. Theory Appl. 180, 117–139 (2019)

    MathSciNet  Article  Google Scholar 

  14. Jeyakumar, V., Yen, N.D.: Solution stability of nonsmooth continuous systems with applications to cone-constrained optimization. SIAM J. Optim. 14, 1106–1127 (2004)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  16. Levy, A.B., Rockafellar, R.T.: Variational conditions and the proto-differentiation of partial subgradient mappings. Nonlinear Anal. 26, 1951–1964 (1996)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    Google Scholar 

  19. Mordukhovich, B.S.: Variational Analysis and Applications. Springer Monographs in Mathematics. Springer, Cham (2018)

    Book  Google Scholar 

  20. Mordukhovich, B.S., Nam, N.M., Nhi, N.T.Y.: Partial second-order subdifferentials in variational analysis and optimization. Numer. Funct. Anal. Optim. 35, 1113–1151 (2014)

    MathSciNet  Article  Google Scholar 

  21. Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. 116, 369–396 (2009)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  23. Mordukhovich, B.S., Outrata, J.V., Ramírez, C.H.: Second-order variational analysis in conic programming with applications to optimality and stability. SIAM J. Optim. 25, 76–101 (2015)

    MathSciNet  Article  Google Scholar 

  24. Mordukhovich, B.S., Outrata, J.V., Sarabi, M.E.: Full stability of locally optimal solutions in second-order cone programs. SIAM J. Optim. 24, 1581–1613 (2014)

    MathSciNet  Article  Google Scholar 

  25. 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  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  27. Mordukhovich, B.S., Sarabi, M.E.: Variational analysis and full stability of optimal solutions to constrained and minimax problems. Nonlinear Anal. 121, 36–53 (2015)

    MathSciNet  Article  Google Scholar 

  28. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  Google Scholar 

  29. Outrata, J.V., Ramírez, C. H.: On the Aubin property of critical points to perturbed second-order cone programs. SIAM J. Optim. 21, 798–823 (2011)

    MathSciNet  Article  Google Scholar 

  30. Polyak, B.T.: Introduction to Optimization, 2nd edn. Translated from the Russian. Optimization Software. Inc. Publications Division, New York (2010)

    Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  32. Robinson, S.M.: Stability theory for systems of inequalities. I. Linear systems. SIAM J. Numer. Anal. 12, 754–769 (1975)

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

  35. Robinson, S.M.: Generalized equations and their solutions II. Applications to nonlinear programming. Math. Programming Stud. 19, 200–221 (1982)

    MathSciNet  Article  Google Scholar 

  36. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

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

    Book  Google Scholar 

  38. Ruszczyński, A.: Nonlinear Optimization. Princeton University Press, Princeton (2006)

    Book  Google Scholar 

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

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The research of the first and the third authors is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED), under grant number 101.01-2018.308. The second author was supported by the Grant MOST 105-2221-E-039-009-MY3 (Taiwan). The authors would like to thank the three anonymous referees for their very careful readings and valuable suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Hanoi Pedagogical University 2, Vinh Phuc, Vietnam

    D. T. K. Huyen

  2. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    J.-C. Yao

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

    N. D. Yen

Authors
  1. D. T. K. Huyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J.-C. Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. D. Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. T. K. Huyen.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huyen, D.T.K., Yao, JC. & Yen, N.D. The Stationary Point Set Map in General Parametric Optimization Problems. Set-Valued Var. Anal 30, 305–327 (2022). https://doi.org/10.1007/s11228-020-00557-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-020-00557-x

Keywords

  • C 2-smooth optimization problem
  • Stationary point set
  • Total perturbation
  • Robinson stability
  • Lipschitz-like property
  • Second-order subdifferential
  • Coderivative

Mathematics Subject Classification (2010)

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