The improvement with relative errors of he et al.'s inexact alternating direction method for monotone variational inequalities

The proximal alternating direction method (PADM) solves a kind of structured monotone variational inequalities (VI) via solving a series of lower-dimensional and strongly monotone sub-VIs. We present a new inexact PADM which solves the involved sub-VIs approximately. The new criteria for solving the involved sub-VIs approximately are relaxed than those presented by He et al. [1]. In addition, the new criteria make use of the relative errors of each iteration instead of the absolute errors to control the accuracy of the inexact solutions of the sub-VIs.