A new alternating direction method for co-coercive variational inequality problems

Abstract

This paper presents a new alternating direction method for solving co-coercive variational inequality problems, where the feasible set is the intersection of a simple set and polyhedron defined by a system of linear equations. The proposed method can be viewed as a combination of Han and Lo’s alternating direction method [D.R. Han, H.K. Lo, A new alternating direction method for a class of nonlinear variational inequality problems, Journal of Optimization Theory and Applications 112 (3) (2002) 549–560] for such class of variational inequality problems and Li, Liao and Yuan’s modified descent method for co-coercive variational inequality problems [M. Li, L.Z. Liao, X.M. Yuan, A modified descent projection method for co-coercive variational inequalities, European Journal of Operational Research 189 (2) (2008) 310–323]. Thus, it possesses the advantages of both Han and Lo’s alternating direction method, which solves a series of small-scale easier problems to solve the original variational inequality problem, and Li, Liao and Yuan’s modified descent method, which is simple provided that the feasible set is simple. We test the new method and compare it with Han and Lo’s method and Li, Liao and Yuan’s modified descent method, and the numerical results show that our new method is suitable for such class of variational inequality problems.