Set-Valued Analysis

, Volume 7, Issue 4, pp 323–345

A Hybrid Approximate Extragradient – Proximal Point Algorithm Using the Enlargement of a Maximal Monotone Operator

  • M. V. Solodov
  • B. F. Svaiter

DOI: 10.1023/A:1008777829180

Cite this article as:
Solodov, M.V. & Svaiter, B.F. Set-Valued Analysis (1999) 7: 323. doi:10.1023/A:1008777829180


We propose a modification of the classical extragradient and proximal point algorithms for finding a zero of a maximal monotone operator in a Hilbert space. At each iteration of the method, an approximate extragradient-type step is performed using information obtained from an approximate solution of a proximal point subproblem. The algorithm is of a hybrid type, as it combines steps of the extragradient and proximal methods. Furthermore, the algorithm uses elements in the enlargement (proposed by Burachik, Iusem and Svaiter) of the operator defining the problem. One of the important features of our approach is that it allows significant relaxation of tolerance requirements imposed on the solution of proximal point subproblems. This yields a more practical proximal-algorithm-based framework. Weak global convergence and local linear rate of convergence are established under suitable assumptions. It is further demonstrated that the modified forward-backward splitting algorithm of Tseng falls within the presented general framework.

maximal monotone operatorenlargement of a maximal monotone operatorproximal point methodextragradient method

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • M. V. Solodov
    • 1
  • B. F. Svaiter
    • 1
  1. 1.Instituto de Matemática Pura e AplicadaEstrada Dona Castorina 110Rio de Janeiro, RJBrazil