Implicit Error Bounds for Picard Iterations on Hilbert Spaces

Vietnam Journal of Mathematics volume 46, pages243–258(2018)Cite this article

Abstract

We investigate the role of error bounds, or metric subregularity, in the convergence of Picard iterations of nonexpansive maps in Hilbert spaces. Our main results show, on one hand, that the existence of an error bound is sufficient for strong convergence and, on the other hand, that an error bound exists on bounded sets for nonexpansive mappings possessing a fixed point whenever the space is finite dimensional. In the Hilbert space setting, we show that a monotonicity property of the distances of the Picard iterations is all that is needed to guarantee the existence of an error bound. The same monotonicity assumption turns out also to guarantee that the distance of Picard iterates to the fixed point set converges to zero. Our results provide a quantitative characterization of strong convergence as well as new criteria for when strong, as opposed to just weak, convergence holds.

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

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

Notes

  1. 1.

    Though it is called the Friedrichs angle, the notion goes back at least to Jordan[18, Eq. 60, pp. 122–130].

References

  1. 1.

    Baillon, J.B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houst. J. Math. 4, 1–9 (1978)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books Math./Ouvrages Math. SMC. Springer, New York (2011)

    Google Scholar 

  4. 4.

    Bauschke, H.H., Deutsch, F., Hundal, H.: Characterizing arbitrarily slow convergence in the method of alternating projections. Int. Trans. Oper. Res. 16, 413–425 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bauschke, H.H., Deutsch, F., Hundal, H., Park, S.-H.: Accelerating the convergence of the method of alternating projections. Trans. Am. Math. Soc. 355, 3433–3461 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Bauschke, H.H., Noll, D., Phan, H.M.: Linear and strong convergence of algorithms involving averaged nonexpansive operators. J. Math. Anal. Appl. 421, 1–20 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Bolte, J., Nguyen, T.P., Peypouquet, J., Suter, B.W.: From error bounds to the complexity of first-order descent methods for convex functions. Math. Program. 165, 471–507 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Borwein, J.M., Li, G., Tam, M.K.: Convergence rate analysis for averaged fixed point iterations in common fixed point problems. SIAM J. Optim. 27, 1–33 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Borwein, J.M., Li, G., Yao, L.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semialgebraic convex sets. SIAM J. Optim. 24, 498–527 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Boţ, R.I., Csetnek, E.R.: A dynamical system associated with the fixed points set of a nonexpansive operator. J. Dyn. Differ. Equ. 29, 155–168 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Drusvyatskiy, D., Ioffe, A.D., Lewis, A.: Transversality and alternating projections for nonconvex sets. Found. Comput. Math. 15, 1637–1651 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Friedrichs, K.: On certain inequalities and characteristic value problems for analytic functions and for functions of two variables. Trans. Am. Math. Soc. 41, 321–364 (1937)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Genel, A., Lindenstrauss, J.: An example concerning fixed points. Isr. J. Math. 22, 81–86 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Hundal, H.: An alternating projection that does not converge in norm. Nonlinear Anal. 57, 35–61 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Ioffe, A.D.: Nonlinear regularity models. Math. Program. Ser. B 139, 223–242 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 59, 65–71 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Jordan, C.: Essai sur la géométrie á n dimensions. Bull. Soc. Math. Fr. 3, 103–174 (1875)

    Article  MATH  Google Scholar 

  19. 19.

    Kayalar, S., Weinert, H.: Error bounds for the method of alternating projections. Math. Control Signals Syst. 1, 43–59 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Leventhal, D.: Metric subregularity and the proximal point method. J. Math. Anal. Appl. 360, 681–688 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Li, G., Nghia, T.T.A., Mordukhovich, B.S., Phạm, T.S.: Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates. Math Program. https://doi.org/10.1007/s10107-016-1014-6 (2016)

  22. 22.

    Luke, D.R., Thao, N.H., Tam, M.K.: Quantitative convergence analysis of iterated expansive, set-valued mappings. Math. Oper. Res. (to appear)

  23. 23.

    Luke, D.R., Thao, N.H., Teboulle, M.: Necessary conditions for linear convergence of Picard iterations and application to alternating projections. arXiv:1704.08926 (2017)

  24. 24.

    Noll, D., Rondepierre, A.: On local convergence of the method of alternating projections. Found. Comput. Math. 16, 425–455 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    von Neumann, J.: Functional Operators, Vol. II. The Geometry of Orthogonal Spaces Annals of Mathematical Studies, vol. 22. Princeton University Press, Princeton (1950)

    Google Scholar 

Download references

Funding

D. Russell Luke was supported in part by German Israeli Foundation Grant G-1253-304.6 and Deutsche Forschungsgemeinschaft Research Training Grant RTG2088.

Nguyen H. Thao was supported by German Israeli Foundation Grant G-1253-304.6 and the European Research Council grant agreement no. 339681.

Matthew K. Tam was supported by Deutsche Forschungsgemeinschaft Research Training Grant 2088 and a Postdoctoral Fellowship from the Alexander von Humboldt Foundation.

Author information

Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, 37083, Göttingen, Germany

    D. Russell Luke & Matthew K. Tam

  2. Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands

    Nguyen H. Thao

Authors
  1. D. Russell Luke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen H. Thao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew K. Tam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Russell Luke.

Additional information

This paper is dedicated to Professor Michel Théra on his 70th birthday.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Luke, D.R., Thao, N.H. & Tam, M.K. Implicit Error Bounds for Picard Iterations on Hilbert Spaces. Vietnam J. Math. 46, 243–258 (2018). https://doi.org/10.1007/s10013-018-0279-x

Download citation

Keywords

  • Averaged operators
  • Error bounds
  • Strong convergence
  • Fixed points
  • Picard iteration
  • Metric regularity
  • Metric subregularity
  • Nonexpansiveness

Mathematics Subject Classification (2010)

  • 49J53
  • 65K10
  • 49M05
  • 49M27
  • 65K05
  • 90C30