Comments on “The Proximal Point Algorithm Revisited”

  • Yunda Dong


Very recently, the author gave an upper bound on a decreasing positive sequence. And, he made use of it to improve a classical result of Brézis and Lions concerning the proximal point algorithm for monotone inclusion in an infinite-dimensional Hilbert space. One assumption is the algorithm’s strong convergence. In this paper, we derive a new upper bound on this decreasing positive sequence and thus achieve the same improvement without requiring this assumption.


Monotone inclusion Convex minimization Proximal point algorithm Rate of convergence 

Mathematics Subject Classification

58E35 65K15 


  1. 1.
    Dong, Y.D.: The proximal point algorithm revisited. J. Optim. Theory Appl. 161, 478–489 (2014)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brézis, H., Lions, P.L.: Produits infinis de resolvantes. Isr. J. Math. 29, 329–345 (1978)MATHCrossRefGoogle Scholar
  3. 3.
    Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley Interscience, New York (1984)MATHGoogle Scholar
  5. 5.
    Martinet, B.: Regularisation d’inéquations variationelles par approximations successives. Rev. Fr. d’Informatique Recherche Opér. 4, 154–158 (1970)MATHMathSciNetGoogle Scholar
  6. 6.
    Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. Fr. 93, 273–299 (1965)MATHMathSciNetGoogle Scholar
  7. 7.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Güler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Zaslavski, A.J.: Maximal monotone operators and the proximal point algorithm in the presence of computational errors. J. Optim. Theory Appl. 150, 20–32 (2011)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

Personalised recommendations