The unified framework of some proximal-based decomposition methods for monotone variational inequalities with separable structure

Some existing decomposition methods for solving a class of variational inequalities (VI) with separable structures are closely related to the classical proximal point algorithm, as their decomposed sub-VIs are regularized by proximal terms. Differing in whether the generated sub-VIs are suitable for parallel computation, these proximal-based methods can be categorized into the parallel decomposition methods and alternating decomposition methods. This paper generalizes these methods and thus presents the unified framework of proximal-based decomposition methods for solving this class of VIs, in both exact and inexact versions. Then, for various special cases of the unified framework, we analyze the respective strategies for fulfilling a condition that ensures the convergence, which are realized by determining appropriate proximal parameters. Moreover, some concrete numerical algorithms for solving this class of VIs are derived. In particular, the inexact version of the unified framework gives rise to some implementable algorithms that allow the involved sub-VIs to be solved under those favorable criteria that are well-developed in the literature of the proximal point algorithm.