Research On Iterative Algorithms For Some Variational Inequalities

In this dissertation, we present a projection gradient algorithm for random variational inequalities and a modified inertial extragradient method for variational inequalities. This dissertation is divided into three chapters.In the first chapter, we introduce the background and the present. situation of the varia.tional inequalities and random variational inequalities. Some prelim-inaries and the main content of this paper are listed.In the second chapter, we propose a projection gradient algorithm for ran-dom variational inequalities in Euclidean space. For this method, we need only one projection onto the set C and only one value of operator F per iteration.These facts make the proposed algorithm very attractive for the cases where the information of the function values of F and the computations of the projection onto C are expensive. Firstly, this chapter introduces some useful definitions and lemmas. Then, we establish the projection gradient algorithm for random vari-ational inequalities. Finally, we prove the globally convergence of the iterative sequence under some suitable assumptions.In the third chapter, we present a modified inertial extragradient algorithm for classical variational inequalities in Euclidean space. Its main work is to intro-duce a linesearch procedure to relax the Lipschitz continuity of F. Firstly, this chapter lists some related definitions and lemmas. Then, we establish the modi-fied inertial extragradient method for variational inequalities. and we prove the globally convergence of the iterative sequence under some suitable assumptions.Finally, the numerical experiments are reported, which shows that the modified inertial extragradient algorithm can speed up the existing extragradient algorith-m , the inertial extragradient algorithm and the double projection algorithm in some situations.