On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds

In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.

An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

Parallel Tseng’s Extragradient Methods for Solving Systems of Variational Inequalities on Hadamard Manifolds

The aim of this article is to study two efficient parallel algorithms for obtaining a solution to a system of monotone variational inequalities (SVI) on Hadamard manifolds. The parallel algorithms are inspired by Tseng’s extragradient techniques with new step sizes, which are established without the knowledge of the Lipschitz constants of the operators and line-search. Under the monotonicity assumptions regarding the underlying vector fields, one proves that the sequences generated by the methods converge to a solution of the monotone SVI whenever it exists.

Iterations for systems of variational inequalities, fixed points of pseudocontractions and zeros of accretive operators

In this paper, we introduce implicit composite three-step Mann iterations for finding a common solution of a general system of variational inequalities, a fixed point problem of a countable family of pseudocontractive mappings and a zero problem of an accretive operator in Banach spaces. Strong convergence of the suggested iterations are given.