Journal of Global Optimization

, Volume 50, Issue 3, pp 531–535 | Cite as

A note on the regularized proximal point algorithm



Recently, Xu (J Glob Optim 36:115–125 (2006)) introduced a regularized proximal point algorithm for approximating a zero of a maximal monotone operator. In this note, we shall prove the strong convergence of this algorithm under some weaker conditions.


Maximal monotone operator Proximal point algorithm Firmly nonexpansive operator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauschke H.H., Combettes P.L.: A weak-to-strong convergence principle for fejér-monotone methods in hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)CrossRefGoogle Scholar
  2. 2.
    Goebel K., Kirk W.A.: Topics on Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  3. 3.
    Güler O.: On the convergence of the proximal point algorithm for convex optimization. SIAM J. Control Optim. 29, 403–419 (1991)CrossRefGoogle Scholar
  4. 4.
    Lehdili N., Moudafi A.: Combining the proximal algorithm and Tikhonov method. Optimization 37, 239–252 (1996)CrossRefGoogle Scholar
  5. 5.
    Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)CrossRefGoogle Scholar
  6. 6.
    Solodov M.V., Svaiter B.F.: Forcing strong convergence of proximal point iterations in a hilbert space. Math. Progr. Ser. A 87, 189–202 (2000)Google Scholar
  7. 7.
    Song Y., Yang C.: A note on a paper a regularization method for the proximal point algorithm. J. Glob. Optim. 43, 171–174 (2009)CrossRefGoogle Scholar
  8. 8.
    Suzuki T.: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 135, 99–106 (2007)CrossRefGoogle Scholar
  9. 9.
    Xu H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)CrossRefGoogle Scholar
  10. 10.
    Xu H.K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36, 115–125 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of MathematicsLuoyang Normal UniversityLuoyangChina

Personalised recommendations