On Theory Of Smoothing Method And Resolvent Operator Method For Variational Inequality Problems And Complementarity Problems

The thesis considers smoothing method for nonlinear complementarity problems (NCP(F)) in Rⁿ, and resolvent operator method for generalized mixed vari-ational inequality problems(GMVI) in Hilbert space, respectively. We propose two improved methods firstly, then their global convergences are proved and the convergence rates are also analyzed. It comprises the following three chapters.Chapter 1 is devoted to the introduction of the thesis, which mainly contains the current development of resolvent operator method and smoothing method, and furthermore, the main task and some indispensable knowledge are also presented briefly.In chapter 2, we study a smoothing method for NCP(F). By this method, we try to give a positive answer to the open problem proposed by B. Chen and N. Xiu (2001). When F(·) is a linear function, B. Chen and N. Xiu (2001) solve the open problem. In this paper, combining the method proposed by B. Chen and N. Xiu (2001) with the infeasible non-interior path following method proposed by S. Xu (2000), we give a new non-interior smoothing method which can solve the above open problem when F(·) is a nonlinear function. Under the condition that F(·) is a P₀-function and R₀-function, and F'(·) is Lipschitz continuous, we prove its global convergence, and furthermore, the global Q-linear convergent rate and local Q-quadratic convergence rate are also proved under the same condition.In chapter 3, we study the resolvent operator method for GMVI. Firstly, we...