Modified Douglas-Rachford Splitting Methods for Structured Inverse Variational Inequality Problems

We give new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to decomposed smaller and easier subproblems. The main computational tasks per iteration are the evaluation of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases that the resolvent operator evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear and linear convergence rate results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods.