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Regularity of Newton’s iteration for general parametric variational system

  • Wei OuyangEmail author
  • Binbin Zhang
Article
  • 92 Downloads

Abstract

In this work, we apply the contraction mapping principle for set-valued mappings to study the Lipschitz-like property of the solution mapping to general parametric variational system obtained via canonical perturbation of a generalized equation. Then, under the assumptions of partial metric regularity and partial Lipschitz-like property, we obtained quadratic convergence of the iterative sequence when started close enough to the solution and provided estimates for the convergence parameters. Upon reconsidering all associated infinite sequences of Newton’s iterates as the image of a mapping defined on a sequence space, we established efficient conditions ensuring such a mapping to possess partial Lipschitz-like properties and also provided estimates of the moduli.

Keywords

Newton’s method parametric variational system (partial) metric regularity (partial) Lipschitz-like property 

Mathematics Subject Classification

49J53 49M37 65J15 90C31 

Notes

Acknowledgements

The authors would like to express their gratitude to Michel Théra for his helpful comments and suggestions. This work was supported by National Natural Science Foundation of China (11801500, 11771384), Yunnan Provincial Science and Technology Department Research Fund (2017FD070), Yunnan Provincial Department of Education Research Fund (2019J0040) and the Fund for Fostering Talents in Kunming University of Science and Technology (KKSY 201807022).

References

  1. 1.
    Adly, S., Cibulka, R., Ngai, H.V.: Newton’s method for solving inclusions using set-valued approximations. SIAM J. Optim. 25, 159–184 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adly, S., Ngai, H.V., Nguyen, V.V.: Newton’s method for solving generalized equations: Kantorovich’s and Smale’s approaches. J. Math. Anal. Appl. 439, 396–418 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Adly, S., Ngai, H.V., Nguyen, V.V.: Stability of metric regularity with set-valued perturbations and application to Newton’s method for solving generalized equations. Set Valued Var. Anal. 25, 543–567 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Aragon Artacho, F.J., Dontchev, A.L., Gaydu, M., Geoffroy, M.H., Veliov, V.M.: Metric regularity of Newton’s iteration. SIAM J. Optim. 49, 339–362 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cibulka, R., Dontchev, A.L., Geoffroy, M.H.: Inexact newton methods and Dennis–Moré theorem for nonsmooth generalized equations. SIAM J. Control Optim. 53, 1003–1019 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)CrossRefGoogle Scholar
  7. 7.
    Dontchev, A.L., Rockafellar, R.T.: Newton’s method for generalized equations: a sequential implicit function theorem. Math. Progr. Ser. B 123, 139–159 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dontchev, A.L., Rockafellar, R.T.: Convergence of inexact Newton methods for generalized equations. Math. Progr. Ser. B 139, 115–137 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  10. 10.
    Ioffe, A.D.: On regularity concepts in variational analysis. J. Fixed Point Theory Appl. 8, 339–363 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Izmailov, A.F., Solodov, M.V.: Newton-Type Methods for Optimization and Variational Problems. Springer, New York (2014)CrossRefGoogle Scholar
  12. 12.
    Lu, S., Robinson, S.M.: Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. 33, 689–711 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I. Basic Theory. II. Applications. Springer, Berlin (2006)CrossRefGoogle Scholar
  14. 14.
    Mordukhovich, B.S., Nghia, T.T.A.: Local monotonicity and full stability for parametric variational systems. SIAM J. Optim. 26, 1032–1059 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mordukhovich, B.S., Nghia, T.T.A., Pham, D.T.: Full stability of general parametric variational systems. Set Valued Var. Anal. 26, 911–946 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ouyang, W., Zhang, B., He, Q.: Lipschitzian stability of fully parameterized generalized equations. Appl. Anal. 97, 1717–1729 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ouyang, W., Zhang, B.: Newton’s method for fully parameterized generalized equations. Optimization 67, 2061–2080 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Robinson, S.M.: Generalized equations and their solutions. I. Basic theory. Point-to-set maps and mathematical programming. Math. Progr. Stud. 10, 128–141 (1979)CrossRefGoogle Scholar
  19. 19.
    Robinson, S.M.: Variational conditions with smooth constraints. Structure and analysis. Math. Progr. 97, 245–265 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of MathematicsYunnan Normal UniversityKunmingPeople’s Republic of China
  2. 2.School of ScienceKunming University of Science and TechnologyKunmingPeople’s Republic of China

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