Abstract:

This paper considers the monotone variational inequality problems with two separable blocks subject to linear coupling constraints. Problems of this type arise in many contemporary applications including traffic assignment and economics. Based on its favorable separable structure, splitting type methods have been studied. In this paper, we introduce a new inexact parallel alternating direction method with a substitution to solve this family of problems. At each iteration, one can get a predictor by using projection in  parallel fashion, then corrects the predictor to generate the new iterate. For the proposed algorithm, we prove its convergence under mild conditions via the analytic framework of contractive type methods. Some numerical results  are reported to support the efficiency of the new method. Moreover,  the proposed method can be extended to solve the variational inequality problems with multi-blocks.

Key words: variational inequalities, alternating direction methods, parallel methods, prediction-correction methods