Skip to main content
SpringerLink
Log in

Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f (x) + ℱ(x) ∋ 0, where X and Y are Banach spaces, and U is an open subset of X, f : U → Y is a nonsmooth function and ℱ : X ⇉ Y is a set-valued mapping with closed graph. We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method. Specifically, under the point-based approximation of f on U and metrically regular property of f + ℱ, we present quadratic rate of convergence of this method. Furthermore, superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f + ℱ is metrically regular. An example of nonsmooth functions that have p-point-based approximation is given. Moreover, a numerical experiment is given which illustrates the theoretical result.

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. Adly S, Cibulka R, Ngai H V. Newton’s method for solving inclusions using set-valued approximations. SIAM J Optim, 2015, 25: 159–184

    Article  MathSciNet  Google Scholar 

  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, 2016, 439: 396–418

    Article  MathSciNet  Google Scholar 

  3. Aragón Artacho F J, Belyakov M, Dontchev A L, et al. Local convergence of quasi-Newton methods under metric regularity. Comput Optim Appl, 2014, 58: 225–247

    Article  MathSciNet  Google Scholar 

  4. Aragón Artacho F J, Dontchev A L, Gaydu M, et al. Metric regularity of Newton’s iteration. SIAM J Control Optim, 2011, 49: 339–362

    Article  MathSciNet  Google Scholar 

  5. Aragón Artacho F J, Geoffroy M H. Uniformity and inexact version of a proximal point method for metrically regular mappings. J Math Anal Appl, 2007, 335: 168–183

    Article  MathSciNet  Google Scholar 

  6. Argyros I K. On a nonsmooth version of Newton’s method based on Hölderian assumptions. Int J Comput Math, 2007, 84: 1747–1756

    Article  MathSciNet  Google Scholar 

  7. Cibulka R, Dontchev A L, Geoffroy M H. Inexact Newton methods and Dennis-Moré theorems for nonsmooth generalized equations. SIAM J Control Optim, 2015, 53: 1003–1019

    Article  MathSciNet  Google Scholar 

  8. Cibulka R, Dontchev A L, Preininger J, et al. Kantorovich-type theorems for generalized equations. J Convex Anal, 2018, 25: 459–486

    MathSciNet  MATH  Google Scholar 

  9. Cibulka R, Preininger J, Roubal T. On uniform regularity and strong regularity. Optimization, 2019, in press

  10. Dontchev A L. Local analysis of a Newton-type method based on partial linearization. Lect Appl Math, 1996, 32: 295–306

    MathSciNet  MATH  Google Scholar 

  11. Dontchev A L. Local convergence of the Newton method for generalized equation. C R Math Acad Sci Paris, 1996, 322: 327–331

    MathSciNet  MATH  Google Scholar 

  12. Dontchev A L. Uniform convergence of the Newton method for Aubin continuous maps. Serdica Math J, 1996, 22: 385–398

    MathSciNet  MATH  Google Scholar 

  13. Dontchev A L, Hager W W. An inverse mapping theorem for set-valued maps. Proc Amer Math Soc, 1994, 121: 481–498

    Article  MathSciNet  Google Scholar 

  14. Dontchev A L, Lewis A S, Rockafellar R T. The radius of metric regularity. Trans Amer Math Soc, 2002, 355: 493–517

    Article  MathSciNet  Google Scholar 

  15. Dontchev A L, Rockafellar R T. Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal, 2004, 12: 79–109

    Article  MathSciNet  Google Scholar 

  16. Dontchev A L, Rockafellar R T. Convergence of inexact Newton methods for generalized equatins. Math Program Ser B, 2013, 139: 115–137

    Article  Google Scholar 

  17. Dontchev A L, Rockafellar R T. Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd ed. New York: Springer, 2014

    MATH  Google Scholar 

  18. Ferreira O P, Silva G N. Inexact Newton’s method to nonlinear functions with values in a cone. ArXiv:1510.01947, 2015

  19. Ferris M C, Pang J S. Engineering and economic applications of complementarity problems. SIAM Rev, 1997, 39: 669–713

    Article  MathSciNet  Google Scholar 

  20. Geoffroy M H, Pietrus A. A general iterative procedure for solving nonsmooth generalized equations. Comput Optim Appl, 2005, 31: 57–67

    Article  MathSciNet  Google Scholar 

  21. Izmailov A F, Solodov M V. Newton-Type Methods for Optimization and Variational Problems. Springer Series in Operations Research and Financial Engineering. Cham: Springer, 2014

    Google Scholar 

  22. Klatte D, Kummer B. Nonsmooth Equations in Optimizations: Reqularity, Calculus, Methods and Applications. Dordrecht-Boston-London: Kluwer Acad Publ, 2002

    MATH  Google Scholar 

  23. Klatte D, Kummer B. Approximations and generalized Newton methods. Math Program, 2018, 168: 673–716

    Article  MathSciNet  Google Scholar 

  24. Mordukhovich B S. Sensitivity analysis in nonsmooth optimization. In: Theoretical Aspects of Industrial Design. Philadelphia: SIAM, 1992, 32–46

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Pietrus A. Does Newton’s method for set-valued maps converges uniformly in mild differentiability context? Rev Colombiana Mat, 2000, 32: 49–56

    MathSciNet  MATH  Google Scholar 

  27. Pietrus A. Non diffrential perturbed Newton’s method for functions with values in a cone. Investigación Oper, 2014, 35: 58–67

    MathSciNet  MATH  Google Scholar 

  28. Qi L, Sun J. A nonsmooth version of Newton’s method. Math Program, 1993, 58: 353–367

    Article  MathSciNet  Google Scholar 

  29. Rashid M H. A convergence analysis of Gauss-Newton-type method for Hoölder continuos maps. Indian J Math, 2015, 57: 181–198

    MathSciNet  Google Scholar 

  30. Rashid M H. Convergence analysis of extended Hummel-Seebeck-type method for solving variational inclusions. Vietnam J Math, 2016, 44: 709–726

    Article  MathSciNet  Google Scholar 

  31. Rashid M H. Extended Newton-type method and its convergence analysis for nonsmooth generalized equations. J Fixed Point Theory Appl, 2017, 19: 1295–1313

    Article  MathSciNet  Google Scholar 

  32. Rashid M H. Convergence analysis of a variant of Newton-type method for generalized equations. Int J Comput Math, 2018, 95: 584–600

    Article  MathSciNet  Google Scholar 

  33. Rashid M H, Sardar A. Convergence of the Newton-type method for generalized equations. Ganit, 2015, 35: 27–40

    MathSciNet  Google Scholar 

  34. Rashid M H, Wang J H, Li C. Convergence analysis of a Gauss-type proximal point method for metrically regualr mappings. J Nonlinear Convex Anal, 2013, 14: 627–635

    MathSciNet  MATH  Google Scholar 

  35. Rashid M H, Yu S H, Li C, et al. Convergence analysis of the Gauss-Newton method for Lipschitz-like mappings. J Optim Theory Appl, 2013, 158: 216–233

    Article  MathSciNet  Google Scholar 

  36. Rashid M H, Yuan Y-X. Convergence properties of a restricted Newton-type method for generalized equations with metrically regular mappings. Appl Anal, 2019, in press

  37. Robinson S M. Generalized equations and their solutions, part I: Basic theory. In: Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol. 10. Berlin-Heidelberg: Springer, 1979, 128–141

    Chapter  Google Scholar 

  38. Robinson S M. Generalized equations and their solutions, part II: Applications to nonlinear programming. In: Optimality and Stability in Mathematical Programming. Mathematical Programming Studies, vol. 19. Berlin-Heidelberg: Springer, 1982, 200–221

    Chapter  Google Scholar 

  39. Robinson S M. Normal maps induced by linear transformations. Math Oper Res, 1992, 17: 691–714

    Article  MathSciNet  Google Scholar 

  40. Robinson S M. Newton’s method for a class of nonsmooth functions. Set-Valued Analysis, 1994, 2: 291–305

    Article  MathSciNet  Google Scholar 

  41. Rockafellar R T, Wets R J-B. Variational Analysis. Berlin: Springer-Verlag, 1997

    MATH  Google Scholar 

  42. Silva G N. Kantorovich’s theorem on Newton’s method for solving generalized equations under the majorant condition. Appl Math Comput, 2016, 286: 178–188

    MathSciNet  MATH  Google Scholar 

  43. Ulbrich M. Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces. Philadelphia: SIAM, 2011

    Book  Google Scholar 

Download references

Acknowledgements

The first author was supported by CAS-President International Fellowship Initiative (PIFI), Chinese Academy of Sciences, Beijing, China. The second author was supported by National Natural Science Foundation of China (Grants Nos. 11688101 and 11331012). The authors thank the anonymous referees for their insightful comments and constructive suggestions, which improved the initial versions of this manuscript.

Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Mohammed Harunor Rashid & Ya-xiang Yuan

  2. Department of Mathematics, Faculty of Science, University of Rajshahi, Rajshahi, 6205, Bangladesh

    Mohammed Harunor Rashid

Authors
  1. Mohammed Harunor Rashid
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ya-xiang Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammed Harunor Rashid.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rashid, M.H., Yuan, Yx. Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations. Sci. China Math. 63, 39–60 (2020). https://doi.org/10.1007/s11425-019-9757-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-9757-0

Keywords

MSC(2010)

Search

Navigation