Two Partial Parallel Splitting Methods For Solving Monotone Variational Inequalities

Convex optimization and variational inequality problems are common problems inoptimization fields. With the development and integration of different disciplines, theyare used more widely. They have become important tools for studying a wide classproblems arising from mathematics, engineering science and management science.Consequently, designing effective algorithm for solving convex optimization andvariational inequality problems is a hot research topic.In this thesis, convex optimization problems with three separable operators arestudied, some first order splitting algorithms based on the variational inequality theoryare proposed. It is organized as follows:Firstly, for convex optimization problem with three constraint variables, combiningthe parallel splitting augmented Lagrangian method and the alternating directionmethod, a partial parallel splitting method is obtained. A descent direction and anappropriate step size along this descent direction is derived. Convergence of the newmethod is proved under mild assumptions, some numerical results demonstrate that thenew method is efficient.Secondly, for monotone variational inequalities with three separable structures, webring in inexact minimization when we solve the subproblems of the monotonevariational inequalities, and the subproblem is simple to have a closed form solution.Combining the alternating directions method and the parallel splitting augmentedLagrangian method, we get an inexact partial parallel alternating direction method.Convergence of the new method is proved under mild assumptiones, some numericalresults demonstrate that the new method is efficient.