Duong Viet Thong

Weak and strong convergence theorems for variational inequality problems

In this paper, we study the weak and strong convergence of two algorithms for solving Lipschitz continuous and monotone variational inequalities. The algorithms are inspired by Tseng’s extragradient method and the viscosity method with Armijo-like step size rule. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

Abstract The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.

Modified subgradient extragradient method for variational inequality problems

In this paper, we introduce an algorithm as combination between the subgradient extragradient method and inertial method for solving variational inequality problems in Hilbert spaces. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. The proposed algorithm can be considered as an improvement of the previously known inertial extragradient method over each computational step. The performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.

New extragradient-like algorithms for strongly pseudomonotone variational inequalities

The paper considers two extragradient-like algorithms for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The projection method is used to design the algorithms which can be computed more easily than the regularized method. The construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the modulus of strong pseudomonotonicity and the Lipschitz constant of the cost operator. Instead of that, the algorithms use variable stepsize sequences which are diminishing and non-summable. The numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with those of several previously known algorithms.

Inertial extragradient algorithms for strongly pseudomonotone variational inequalities

Abstract The purpose of this paper is to study and analyze two different kinds of inertial type iterative methods for solving variational inequality problems involving strongly pseudomonotone and Lipschitz continuous operators in Hilbert spaces. The projection method is used to design the algorithms which can be computed more easily. The construction of solution approximations and the proof of convergence of the algorithms are performed without prior knowledge of the modulus of strong pseudomonotonicity and the Lipschitz constant of cost operator. Instead of that, the algorithms use variable stepsize sequences which are diminishing and non-summable. The numerical behaviors of the proposed algorithms on a test problem are illustrated and compared with several previously known algorithms.

Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems

In this paper, basing on the subgradient extragradient method and inertial method with line-search process, we introduce two new algorithms for finding a common element of the solution set of a variational inequality and the fixed point set of a quasi-nonexpansive mapping with a demiclosedness property. The weak convergence of the algorithms are established under standard assumptions imposed on cost operators. The proposed algorithms can be considered as an improvement of the previously known inertial extragradient method over each computational step. Finally, for supporting the convergence of the proposed algorithms, we also consider several preliminary numerical experiments on a test problem.

A new projection method for a class of variational inequalities

AbstractIn this paper, we revisit the numerical approach to variational inequality problems involving strongly monotone and Lipschitz continuous operators by a variant of projected reflected gradie...Abstract In this paper, we revisit the numerical approach to variational inequality problems involving strongly monotone and Lipschitz continuous operators by a variant of projected reflected gradient method. Contrary to what done so far, the resulting algorithm uses a new simple stepsize sequence which is diminishing and nonsummable. This brings the main advantages of the algorithm where the construction of aproximation solutions and the formulation of convergence are done without the prior knowledge of the Lipschitz and strongly monotone constants of cost operators. The assumptions in the formulation of theorem of convergence are also discussed in this paper. Numerical results are reported to illustrate the behavior of the new algorithm and also to compare with others.