LQP Alternating Direction Methods For Solving Variational Inequalities With Separable Structure

Variational inequality has a wide range of applications, and it is a very important Tool in the research of optimization. Variational inequality has been used to address many convex optimization problems in the field of information technology which includes image restoration, signal processing, management science, statistical computing, machine learning and so on. It is very convenient to do research on methods of calculating the convex optimization problems by the application of variational inequality.Logarithmic-quadratic proximal(LQP) Alternating direction method is an efficient algorithm for solving variational inequality with separable structure which is a special form of variational inequalities. It is an improved algorithm for original alternating direction method, which not only takes full advantage of the separability of the original problem, and divides the original problem into two relatively low-dimensional sub-problems, but also transforms the two sub-problems into nonlinear equations which will be easier to solve by introducing LQP regularization. This method avoids solving monotone variational problems of original alternating direction method.In this article, we further study LQP alternating direction method and its improved algorithms. The main research work is as follows:Firstly, we propose a new descending LQP alternating direction method by constructing a new descent direction. During the analysis of the algorithm, we present a way of selecting optimal step, and then we prove the global convergence of the algorithm under a weaker assumption. Thus Some numerical results show that the new method is effective in practice.Secondly, we propose a new inexact LQP generalized alternating direction method by combining the generalized alternating direction method. By using this new algorithm, we only need to solve two sub-problems approximately rather than precisely. For the derived algorithm, we prove its global convergence and establish its worst-case convergence rate in the ergodic senses and non-ergodic senses under an appropriate inexactness criterion to illustrate the effectiveness of the new algorithm.