Open Access

An Inexact Proximal-Type Method for the Generalized Variational Inequality in Banach Spaces

Journal of Inequalities and Applications20082007:078124

Received: 27 July 2007

Accepted: 31 October 2007

Published: 31 January 2008


We investigate an inexact proximal-type method, applied to the generalized variational inequality problem with maximal monotone operator in reflexive Banach spaces. Solodov and Svaiter (2000) first introduced a new proximal-type method for generating a strongly convergent sequence to the zero of maximal monotone operator in Hilbert spaces, and subsequently Kamimura and Takahashi (2003) extended Solodov and Svaiter algorithm and strong convergence result to the setting of uniformly convex and uniformly smooth Banach spaces. In this paper Kamimura and Takahashi's algorithm is extended to develop a generic inexact proximal point algorithm, and their convergence analysis is extended to develop a generic convergence analysis which unifies a wide class of proximal-type methods applied to finding the zeroes of maximal monotone operators in the setting of Hilbert spaces or Banach spaces.


Authors’ Affiliations

Department of Mathematics, Shanghai Normal University
Department of Mathematics, University of Pisa
Department of Applied Mathematics, National Sun Yat-Sen University


  1. Burachik RS, Lopes JO, Svaiter BF: An outer approximation method for the variational inequality problem. SIAM Journal on Control and Optimization 2005,43(6):2071–2088. 10.1137/S0363012902415487MathSciNetView ArticleMATHGoogle Scholar
  2. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2003,13(3):938–945.MathSciNetView ArticleMATHGoogle Scholar
  3. Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Mathematical Programming 2000,87(1 (A)):189–202.MathSciNetMATHGoogle Scholar
  4. Bruck RE Jr.: An iterative solution of a variational inequality for certain monotone operators in Hilbert space. Bulletin of the American Mathematical Society 1975,81(5):890–892. 10.1090/S0002-9904-1975-13874-2MathSciNetView ArticleMATHGoogle Scholar
  5. Bruck RE Jr.: Corrigendum: "An iterative solution of a variational inequality for certain monotone operators in Hilbert space". Bulletin of the American Mathematical Society 1976,82(2):353.MathSciNetMATHGoogle Scholar
  6. Iusem AN: On some properties of paramonotone operators. Journal of Convex Analysis 1998,5(2):269–278.MathSciNetMATHGoogle Scholar
  7. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In Nonlinear Functional Analysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968). American Mathematical Society, Providence, RI, USA; 1976:1–308.Google Scholar
  8. Karamardian S: Complementarity problems over cones with monotone and pseudomonotone maps. Journal of Optimization Theory and Applications 1976,18(4):445–454. 10.1007/BF00932654MathSciNetView ArticleMATHGoogle Scholar
  9. Pascali D, Sburlan S: Nonlinear Mappings of Monotone Type. Editura Academiei, Bucharest, Romania; 1978.View ArticleMATHGoogle Scholar
  10. Lions P-L: Une méthode itérative de résolution d'une inéquation variationnelle. Israel Journal of Mathematics 1978,31(2):204–208. 10.1007/BF02760552MathSciNetView ArticleMATHGoogle Scholar
  11. Rockafellar RT: Characterization of the subdifferentials of convex functions. Pacific Journal of Mathematics 1966, 17: 497–510.MathSciNetView ArticleMATHGoogle Scholar


© L. C. Ceng et al. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.