Some New Methods For Variational Inequalities And Complementarity Problems

Some new efficient methods for solving some variational inequality problems and complementarity problems are constructed and analyzed in this paper. Variational inequality problems and complementarity problems can be met in the mathematical modelling of various applications in physics, mechanics, economics, operations research, optimal control theory and traffic assignment problems, e.t.c. It's significant to establish some efficient numerical methods to solving these problems. There have been lots of researches on the solution of these problems, which presented feasible and essential techniques.Parallel computing methods are efficient for large scale science and engineering problems. One of the advantages of synchronous parallel methods is that each subdomain can be handled by a different processor of a parallel computer, which results in high computation efficiency. But, at the synchronization points, the synchronous methods need to wait the fresh information from the other processors. As a result, parallel methods, especially asynchronous parallel methods, become more and more important in scientific and engineering calculations. In this paper we mainly consider asynchronous parallel methods, i.e., methods in which the sub-problem in each processor is solved anew with whatever information is available at the moment, without waiting for new information from all other processors. The asynchronous versions converge faster than the synchronous ones. As an important theoty tool for discussion how to solving the linear or nonlinear equations in parallel, one of the advantages of the multisplitting mehtods is that each processor computes only those entries of the vector, which correspond to the nonzero diagonal entries of the weighted matrice at each major stage of iteration. That is, in some sense, we can separate the large problem into some small subproblems to calculate. As an extension of the multisplitting methods, the weighted additive Schwarz methods (multisplitting Schwarz method) have better performance than the multisplitting methods. The main idea of the so-called nonstationary multi-splitting methods is that at each iteration each processor solves a corresponding subproblem by finite times, using in each time the new calculated vector as a new initial value. Numerical experiments show that the nonstationary models have better performance than the standard (stationary) multisplitting methods in mul-tiprocessors, if a good load balancing among the processors is obtained. In this paper, we extend these methods to solve the complemenarity problems, including linear complementarity problems, mildly nonlinear complementarity problems and affine two-sided obstacle problems. Prom M-matrix to H-matrix, the coefficient matrix of the complementarity problems, we introduce a series of synchronous or asychronous multisplitting Schwarz methods and nonstationary multisplitting methods, and study the convergence of these methods. Numerical experiments show that they are efficient and robust.Domain decomposition method and multi-grid method are two important methods for solving variational inequality problems. Firstly, we proposed a two-level Schwarz method for solving variational inequality problems with nonlinear source terms, which based on active set strategy, the computational domain can be partitioned into subdomains with linear and obstacle-type subproblems. By using this domain decomposition and fast linear solvers, the corresponding subproblems are solved. Secondly, we study a generalized Schwarz method for obstacle problems with a T-monotone operator. As to this method, the subproblems are coupled by Robin condition with a parameter ωi. Numerical experiments show that it is much faster than the classical Schwarz algorithm with an appropriate choice of the parameter. We proved the convergence for the algorithms. Recently, a new method, called Cascading multigrid method, is developed. This kind of method requires no coarse grid corrections at all. Another distinctive feature is that more iterations on coarser levels...