The Projection-type Methods Research For Two Classes Of Variational Inequalities

Projection-type method is one of the most important methods for solving variational inequality problems. In this dissertation, we mainly study the projection-type method for solving two classes of variational inequalities. The full text is divided into four chapters. We summarize as follows:In the first chapter, we introduced the research background and basic definitions and concepts herein.In the second chapter, a new projection method for mixed variational inequalities is introduced in Euclidean spaces. the Armijo-type linesearch is similar to that of He's method for variational inequalities. Under some suitable assumptions, we prove that the sequence generated by the proposed method is globally convergent to a solution of the problem. If,in addition, a certain error bound holds, we analyze the convergence rate of the iterative sequence.In the third chapter, a new projection-type method for generalized variational inequalities is introduced in Euclidean spaces. Under the assumption that the dual variational inequality has a solution, we show that the proposed method is well-defined and prove that the sequence generated by the proposed method is convergent to a solution, where the condition is strictly weaker than the pseudomonotonicity of the mapping used by some authors. We provide an example to support our results. Compared with the recent works of Li and He, and Fang and He, condition(A3) is removed.In the last chapter, a projection-type algorithm for mixed variational inequalities is proposed in Banach spaces. The existence of the solutions of mixed variational inequalities can be verified through the behavior of the sequence generated by the proposed algorithm. If the solution set is nonempty, then the sequence generated by the algorithm is strongly convergent to the solution, which is closest to the initial point in the sense of Bregman distance.Otherwise the sequence is divergent.