Research On Projection Algorithm For Solving Variational Inequalities And Fixed Point Problems

Variational inequality is one of the important components of optimization theo-ry.It is widely emerged in different fields such as operational research,economics and manage-ment science.In order to find an approximate solution of variational inequality problems,many iterative algorithms have been proposed.Projection algorithm is one of the important methods for solving variational inequality problems.In this thesis,incremental constraint projection methods for variational inequality problems have been researched in Rn and the projection methods for finding the comment solution of variational inequality problems and fixed point problems have been researched in H.This thesis consists of six chapters,the main contents are in the following:In Chapter 1,we state the research background of variational inequality problems and fixed point problems,and the research progress of projection algorithm.Then the research motivation and the summary of the main work of this thesis are given.In Chapter 2,we recall some basic concepts and results.In Chapter 3,we study the incremental constraint projection algorithm for variational in-equality problems in Rn(containing random projection algorithm and cyclic projection algorith-m).In our method,we only need twice projections onto a specific half-space at each iteration.Under the assumption that the mapping is monotone plus and Lipschitz continuous,we prove the sequence generated by our method is convergent to a solution of variational inequality problems in almost sure sense.The numerical results illustrate that the proposed algorithm is effective.In Chapter 4,we study the inertial incremental constraint projection algorithm for varia-tional inequality problems in Rn(containing inertial random projection algorithm and inertial cyclic projection algorithm).In our method,we only need once projection onto a specific half-space at each iteration.Under the assumption that the mapping is strongly monotone without Lipschitz continuous,we prove the sequence generated by our method is convergent to a so-lution of the variational inequality problems in almost sure sense.The numerical results show that the proposed algorithm is effective.In Chapter 5,we present a projection algorithm for finding a common solution of varia-tional inequality problems and nonexpansive fixed point problems in H.Under the assumption that the mapping F is monotone uniformly continuous,U is nonexpansive and the solution set is nonempty,we prove that the sequence generated by the proposed method converges strongly to a solution of variational inequality problems and fixed point problems.In Chapter 6,summarizes the main results and innovation of the full text,and further research on variational inequality is expected.