Decomposition Method for a Class of Monotone Variational Inequality Problems

Abstract

In the solution of the monotone variational inequality problem VI(Ω, F), with

$$u = \left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right],Fu = \left[ {\begin{array}{*{20}c} {fx - ATy} \\ {Ax - b} \\ \end{array} } \right],\Omega = \mathcal{X} \times \mathcal{Y},$$

the augmented Lagrangian method (a decomposition method) is advantageous and effective when $\mathcal{X} = \mathcal{R}^m$. For some problems of interest, where both the constraint sets $\mathcal{X}$ and $\mathcal{Y}$ are proper subsets in $\mathcal{R}^n$ and $\mathcal{R}^m$, the original augmented Lagrangian method is no longer applicable. For this class of variational inequality problems, we introduce a decomposition method and prove its convergence. Promising numerical results are presented, indicating the effectiveness of the proposed method.