Alternating Direction Method For Solving A Class Of Monotone Variational Inequalities

Over the past decades, the theory and algorithms of finite-dimensional variational inequality (including complementarity problems) has been developed rapidly and applied broadly to transportation planning, socio-economic analysis, energy modeling, game theory and so on. Particularly the algorithms of finite-dimensional variational inequality have been a very important research subject in computable mathematics. This paper provides some researches on the alternating direction method for solving a class of asymmetric monotone variational inequalities.Alternating direction method is efficient for solving a class of variational inequalities with linear equalities or linear inequalities constraints. The basic idea of the method is to approximate the solution of variational inequalities via solving alternately a linear variational inequality with simple constraints and a well-conditioned system of nonlinear equations. A remarkable merit of the alternating direction method is that it is easy to solve sub-problems. In addition, the sub-problems in the alternating direction method can be solved by many existing efficient mathematical algorithms. The paper generalizes and improves the traditional alternating direction method on the following aspects:1. The traditional alternating direction method was applied to respectively solve a class of variational inequalities with linear equalities or linear inequalities constraints. The method of the paper is applied to solve a class of variational inequalities with both linear equalities and linear inequalities constraints. We have proved the convergence of the method.2. We propose two classes of inexact direction methods and allow to inexactly solving the two sub-problems. We have proved the convergence of the method under mild assumptions.3. We propose self-adaptive alternating direction method. Although the solution of VI(K,f) is invariant under multiplying f by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is aconstant in the original alternating direction. In general, it is difficult to choose a proper parameter β for individual problems. Thus we propose...