Solving A Class Of Monotone Variational Inequalities By A Alternating Direction Method

From 1960's, the theory and algorithms of finite-dimensional variational inequalitieshave been developed rapidly and applied broadly to the theory of economic equilibrium,transportation planning, social-economic analysis and so on. Particulary the algorithmsof finite-dimensional variational inequality have been a very important research field incomputational mathematics. This thesis presents some works in the alternating directionmethod for solving a class of asymmetric monotone variational inequalities. Alternatingdirection method is e?cient for solving a class of variational inequality with linear equalityor linear inequality constraints. The basic idea of the method is to approximate thesolution of variational inequalities via solving alternately two kinds of sub-problems. Thistraditional methods needs alternately solving a linear variational inequality with simpleconstraints and a well-conditioned system of nonlinear equations, and the sub-problemscan be solved easily by many e?cient mathematical algorithms. The thesis generalizesand improves the traditional alternating direction method on the following aspects:1. On the basis of the traditional method, we propose an inexact self-adaptivealternating direction method. The new method allows one kind of sub-problems to besolved inexactly. We have proved the convergence of the method .2. We propose a new alternating direction method and prove the convergence ofthe new method under mild assumptions. This method allows to inexactly solve the twokinds of sub-problems.3. We propose new self-adaptive alternating direction method. The numerical ex-periment has shown that the number of iterations depends significantly on the positiveparameterα,β. But, in general, it is di?cult to choose a proper parameterα,β. Thuswe propose new self-adaptive alternating direction method which adjusts the scale pa-rameter automatically per iteration based on the message of the iterates. We have provedthe convergence of the method under mild assumptions.Finally, we present some numerical results. These numerical tests show that the inex-act self-adaptive alternating direction method and the new alternating direction methodare e?cient.