A New Implementable Prediction-Correction Method for Monotone Variational Inequalities with Separable Structure

Feng Ma,Zhanke Yu,Mingfang Ni

doi:10.1155/2013/941861

Abstract

The monotone variational inequalities capture various concrete applications arising in many areas. In this paper, we develop a new prediction-correction method for monotone variational inequalities with separable structure. The new method can be easily implementable, and the main computational effort in each iteration of the method is to evaluate the proximal mappings of the involved operators. At each iteration, the algorithm also allows the involved subvariational inequalities to be solved in parallel. We establish the global convergence of the proposed method. Preliminary numerical results show that the new method can be competitive with Chen's proximal-based decomposition method in Chen and Teboulle (1994).

Highlights

IntroductionThe variational inequality (VI (Ω, F)) in the finite-dimensional space is to determine a vector u ∈ Ω such that
The variational inequality (VI (Ω, F)) in the finite-dimensional space is to determine a vector u ∈ Ω such that⟨u󸀠 − u, F (u)⟩ ≥ 0, ∀u󸀠 ∈ Ω, (1)where Ω ∈ Rn is a nonempty closed convex subset and F is a continuous mapping from Rn into itself
We develop a new prediction-correction method for monotone variational inequalities with separable structure

Summary

Introduction

The variational inequality (VI (Ω, F)) in the finite-dimensional space is to determine a vector u ∈ Ω such that. (1/β)f)−1(Aυ) and (BTB + (1/β)g)−1(Bυ) could be costly To overcome this difficulty, we propose a new implementable prediction-correction method for the SVI. Han’s method is based on logarithmic-quadratic functions and combined with self-adaptive strategy He [14] presented a parallel splitting augmented Lagrangian method which can be extended to solve the system of equilibrium problems with three separable operators. Yuan and Li [17] developed a logarithmic-quadraticproximal- (LQP-) based decomposition method by applying the LQP terms to regularize the ADM subproblems; Bnouhachem et al [18] studied a new inexact LQP alternating direction method by solving a series of related systems of nonlinear equations.

Preliminaries

The New Algorithm

Description of the Algorithm

Numerical Experiments

Conclusions