Skip to main content
SpringerLink
Log in

The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

It is known, by Rockafellar (SIAM J Control Optim 14:877–898, 1976), that the proximal point algorithm (PPA) converges weakly to a zero of a maximal monotone operator in a Hilbert space, but it fails to converge strongly. Lehdili and Moudafi (Optimization 37:239–252, 1996) introduced the new prox-Tikhonov regularization method for PPA to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in the same space setting. In this paper, the prox-Tikhonov regularization method for the proximal point algorithm of finding a zero for an accretive operator in the framework of Banach space is proposed. Conditions which guarantee the strong convergence of this algorithm to a particular element of the solution set is provided. An inexact variant of this method with error sequence is also discussed.

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.

Similar content being viewed by others

References

  1. Agarwal R.P., O’Regan D., Sahu D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications, Series: Topological Fixed Point Theory and its Applications. Springer, New York (2009)

    Google Scholar 

  2. Alber, Y.I., Butnariu, D., Kassay, G.: Convergence and stability of a regularization method for maximal monotone inclusions and its application to convex optimization, variational analysis and applications. In: Giannessi F., et al. (eds.) Proc. of the 38th Conference of the School of Mathematica G. Stampacchia in Memory of G. Stampacchia and J.-L. Lions, Erice, Italy, June 20–July 1, 2003. Springer, New York, Nonconvex Optimization and its Applications, vol. 79, pp. 89–132 (2005)

  3. Attouch H., Wets R.J.B.: Isometries for Legendre-Fenchel transform. Technical report. Trans. Am. Math. Soc. 296, 33–60 (1986)

    Article  Google Scholar 

  4. Barbu, V., Precupanu, Th.: Convexity and Optimization in Banach spaces, Editura Academiei R. S. R., Bucharest (1978)

  5. Butnariu D., Kassay G.: A proximal-projection method for finding zeros of set-valued operators. SIAM J. Control Optim. 47, 2096–2136 (2008)

    Article  Google Scholar 

  6. Cioranescu I.: Geometry of Banach Spaces, Duality Mapping and Nonlinear Problems. Kluwer, Amsterdam (1990)

    Book  Google Scholar 

  7. Eckstein J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program. 83, 113–123 (1998)

    Google Scholar 

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

    Article  Google Scholar 

  9. Goebel K., Reich S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker Inc, New York (1984)

    Google Scholar 

  10. Ha K.S., Jung J.S.: Strong convergence theorems for accretive operators in Banach space. J. Math. Anal. Appl. 147, 330–339 (1990)

    Article  Google Scholar 

  11. Lehdili N., Moudafi A.: Combining the proximal algorithm and Tikhonov regularization. Optimization 37, 239–252 (1996)

    Article  Google Scholar 

  12. Martinet B.: Regularisation dinequations variationnelles par approximations successives. Rev. FranMcaise Informat. Recherche Operationnelle 4, 154–158 (1970)

    Google Scholar 

  13. Minty G.J.: On the maximal domain of a ‘monotone’ function. Michigan Math. J. 8, 135–137 (1961)

    Article  Google Scholar 

  14. Rockafellar R.T.: Characterization of the subdifferentials of convex functions. Pac. J. Math. 17, 497–510 (1966)

    Google Scholar 

  15. Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  Google Scholar 

  16. Song Y., Yang C.: A note on a paper “A regularization method for the proximal point algorithm”. J. Glob. Optim. 43, 171–174 (2009)

    Article  Google Scholar 

  17. Takahashi W.: Nonlinear Functional Analysis, Fixed Point Theory and its Applications. Yokohama Publishers, Yokohama (2000)

    Google Scholar 

  18. Takahashi W.: Viscosity approximation methods for resolvents of accretive operators in Banach spaces. J. Fixed Point Theory Appl. 1, 135–147 (2007)

    Article  Google Scholar 

  19. Tikhonov A.N.: Solution of incorrectly formulated problems and a regularization method. Soviet Math. Doklady 4, 1035–1038 (1963)

    Google Scholar 

  20. Tikhonov A.N.: Improper problems of optimal planning and stable methods of their solution. Soviet Math. Doklady 6, 1264–1267 (1965)

    Google Scholar 

  21. Tikhonov A.N., Arsenin V.J.: Methods for Solving Ill-Posed Problems. Wiley, New York (1977)

    Google Scholar 

  22. Tossings P.: The perturbed proximal point algorithm and some of its applications. Appl. Math. Optim. 29, 125–159 (1994)

    Article  Google Scholar 

  23. Wong N.C., Sahu D.R., Yao J.C.: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal. 69, 4732–4753 (2008)

    Article  Google Scholar 

  24. Xu H.K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115–125 (2006)

    Article  Google Scholar 

  25. Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India

    D. R. Sahu

  2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    J. C. Yao

Authors
  1. D. R. Sahu
    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

Corresponding author

Correspondence to J. C. Yao.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sahu, D.R., Yao, J.C. The prox-Tikhonov regularization method for the proximal point algorithm in Banach spaces. J Glob Optim 51, 641–655 (2011). https://doi.org/10.1007/s10898-011-9647-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9647-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation