Linearized Alternating Direction Contraction Method

In this paper, we introduce an improved alternating direction method by letting in lin-earization technique. Numerical results show that this method works well under some certain conditions, the CPU time and number of iterations can be significantly reduced. The paper is organized as follows:In Chapter1, we briefly intruduce the application background of variational in-equality, its development in recent years, and the main job of this paper.Chapter2includes some basic knowledge of variational inequality, including the properties of projection, the concept of monotonic operator, the equivalence between projection equation and variational inequality, and yielded three basic inequalities. Some mild assumptions are also imposed for fuuther discussions.In Chapter3, we propose the linearized alternating direction method and a self-adaptive step size rule.In Chapter4, the convergence proofs and convergence rate analysis are given. The convergence result of linearized alternating direction method is derived, then the con-vergence result and O(1/t) convergence rate of linearized alternating direction method with self-adaptive step size are established.In Chapter5, the numerical experimental results on two types of problems are reported, and the efficiency of the proposed algorithms is confirmed. Finally, some conclusions are drawn in the last Chapter.