The Research On Methods For Several Kinds Of Variational Inequality And Complementarity Problems

Variational inequality and complementarity problems have a lot of applications in many fields, such as mechanism, engineering, physics, finance, optimal control theory mathematical models and equilibrium models arised from traffic transportation. Hence, it is meaningful to study the efficient numerical methods for solving variational inequality and complementarity problem. In the past decades, great progress has been made in the study of numerical algorithms and this kind of research emerges in endlessly. In this thesis, we construct and analyze some efficient numerical methods for solving variational inequality and complementarity problem.Many problems arised in science and engineering are often large scale, and require high precision. Hence, we need to design some new and more efficient algorithms for such problems. With the use of parallel computer, it is a very important way to solve those problems in parallel. Multisplitting method is an important theoretical tool for solving linear or nonlinear equations in parallel. Its feature is to decompose a large scale problem into several subproblems and solve those subproblems in parallel. In Chapter 2, we extend this method for solving symmetrical affine second order cone complementarity problem (SOCCP). The multisplitting method we constructed for SOCCP exploits particular features of matrices such as the sparsity and the block structure. In each iteration of the method, each processor deals with a subproblem obtained from one matrix splitting respectively. After that, the results from each processor are summed with weight as the initial value for the next iteration. Moreover, we consider an MAOR-like method to solve the subproblems. Numerical results showed that multisplitting is effective for solving symmetrical affine SOCCP.Domain decomposition method was developed in 1980s. Its main point is to divide the domain into several small subdomains, and to solve the subproblems in related subdomains. It is a very important numerical method for solving variational inequality and complementarity problem. In Chapter 3, we develop and analyze a two-level additive Schwarz method for nonlinear complementarity problem (NCP) with an M-function. Based on certain criterion, this method divides the domain into two subdomains which contain obstacle subproblem and nonlinear equations subproblem, respectively. The advantage of this method is that it offers the possibility of making use of fast nonlinear solvers for the genuinely nonlinear obstacle problems. We prove that this method converges in finite number of iterations. Numerical results show that the method is efficient.According to different optimality conditions of the solution, we study different active set strategies for unilateral and bilateral obstacle problems with a T-monotone operator in Chapter 4. Based on certain criterions, active set strategies can decompose the index set into active set and inactive set, then solve a reduced nonlinear system in inactive set. Contrary to the PSOR and Schwarz methods, no additional linear or nonlinear subproblem solvers are needed. In numerical experiments, we compare active set strategies with PSOR and Schwarz method, and numerical results indicate that active set strategies are valid.In recent years, variational inequality has been extended to various directions, such as generalized variational inequality, mixed variational inequality, generalized variational-like inequality and so on. One of the most important extension is system of variational inequalities. In Chapter 5, we consider a system of generalized variational-like inequalities problems (SGVLIP). Firstly, some approximate problems related to SGVLIP are proposed. The existence theorem of solution of approximate problems is obtained. Then algorithms based on these approximate problems are constructed for the SGVLIP and the convergence of the algorithms are also discussed. In Chapter 6, we study a kind of system of nonlinear variational inequalities (SNVI) and its related auxiliary problems in real Hilbert space. Existence theorems for SNVI and the auxiliary problems are established. Furthermore, by exploiting existence theorem, algorithms for SNVI are constructed and the convergence of the algorithm is discussed. In Chapter 7, we consider the numerical solution of system of mixed nonlinear variational inequalities (SMNVI) in Banach space. The definition ofη—proximal mapping for a proper subdifferentiable functional is introduced. Using the properties ofη—proximal mapping, some iterative algorithms for solving SMNVI are constructed and convergence theorem is also established.