This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar’s proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar’s PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm is a (large-ste...

In this paper, we consider and analyze a new extragradient method for solving pseudomonotone variational inequalities. The new method converges for pseudomonotone Lipschitz continuous operators, which is a weaker condition than monotonicity. The new iterative method includes the extragradient method as a special case. Our proof of convergence is very simple as compared with other methods. Mathe...

In this paper, we introduce and analyze a hybrid steepest-descent extragradient algorithm for solving triple hierarchical pseudomonotone variational inequalities in a real Hilbert space. The proposed algorithm is based on Korpelevich’s extragradient method, Mann’s iteration method, hybrid steepest-descent method and Halpern’s iteration method. Under mild conditions, the strong convergence of th...

In this paper, we propose a new version of extragradient method for the variational inequality problem. The method uses a new searching direction which differs from any one in existing projection-type methods, and is of a better stepsize rule. Under a certain generalized monotonicity condition, it is proved to be globally convergent. @ 2001 Eisevier Science Ltd. All rights reserved. Keywords-Vs...

The purpose of this paper is to introduce a new hybrid extragradient iterative method for finding a common element of the set of points satisfying the Ky Fan inequalities, variational inequality and the set of fixed points of a strict pseudocontraction mapping in Hilbert spaces. Consequently, we obtain the strong convergence of an iterative algorithm generated by the hybrid extragradient projec...

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce an extragradient-like approximation method which is based on so-called extragradient method and viscosity approximation method. We establish...

In this paper we analyze the iteration-complexity of the hybrid proximal extragradient (HPE) method for finding a zero of a maximal monotone operator recently proposed by Solodov and Svaiter. One of the key points of our analysis is the use of a new termination criteria based on the εenlargement of a maximal monotone operator. The advantages of using this termination criterion it that its defin...

In this paper we introduce a hybrid relaxed-extragradient method for finding a common element of the set of common fixed points of N nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The hybrid relaxed-extragradient method is based on two well-known methods: hybrid and extragradient. We derive a strong convergence ...

We propose a hybrid extragradient method for finding a common element of the solution set of a variational inequality problem, the solution set of a general system of variational inequalities, and the fixed-point set of a strictly pseudocontractive mapping in a real Hilbert space. Our hybrid method is based on the well-known extragradient method, viscosity approximation method, and Mann-type it...

Recently, Nemirovski’s analysis indicates that the extragradient method has the O(1/t) convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, we have developed a class of Fejér monotone projection and contraction methods. Until now, only convergence results are available to these projection and contraction methods,...