Partial Parallel Splitting LQP Alternating Direction Methods For Solving A Class Of Separable Structured Variational Inequalities Problems

Variational inequality is a very important tool in the research of optimization,and it has a wide range of applications.We can describe equilibrium problems in economy,many problems in statistics and machine learning,transportation problems,convex programming problems and so on by variational inequalities.In rescent years,it is an important research focus in the field of optimization to study a class of variational inequalities with separable structures.In this paper,we mainly study separable structured variational inequality problem with three operators,The obtained concrete results as follows:Firstly,for separable structured variational inequality problem with three operators,we combine the augmented Lagrangian method and the Logarithmic quadratic proximal(LQP)alternating direction method,and obtain a partial parallel splitting LQP alternating direction methods.We construct two descent directions.By combined with the two descent directions,we get the new direction.And we derive an appropriate step size along this new descent direction.Then,we prove the global convergence of the algorithm under a weaker assumption.Secondly,for separable structured variational inequality problem with three operators,we consider that the cost of accurately solving the sub-problem is too big or not too reality and the inexactly solving the sub-problem is more rapid and effective.Based on partial parallel splitting LQP alternating direction methods,we add an inexact term to solve them inexactly instead of solving the sub-problems exactly.Then,we use them to approximate the sub-problems' real solutions.We obtain a inexact partial parallel splitting LQP alternating direction methods.We construct two descent directions.By combined with the two descent directions,we get the new direction.And we derive an appropriate step size along this new descent direction.We prove the global convergence of the new algorithm under some inexact conditions,and establish its worst-case tO)1(convergence rate in the ergodic sense to illustrate the effectiveness of the new algorithm.