Modified Projection Algorithm For Solving Monotone And Lipschitz Continuous Variational Inequality Problems

In this dissertation,we further studies the existing projection algorithms for solving monotone and Lipschitz continuous variational inequality problems.Firstly,on the basis of the modified subgradient extragradient algorithm proposed by Malitsky and Semenov.We combine the new step-size rule with the inertial method to propose a new modified subgradient extragradient algorithm.The new algorithm does not need to know the Lipschitz modulus of mapping and improves the descending direction of the original algorithm,and then combines with the inertial method to accelerate the convergence of the original algorithm.Moreover,the algorithm needs only one value of underlying mapping and one projection to the feasible set per iteration.Next,on the basis of the inertial two-subgradient extragradient algorithm proposed by Cao and Guo,we combine the viscosity method and the new step-size of inertial method to propose a new inertial two-subgradient extragradient algorithm when the feasible set is a lower level set of a smooth and convex function.Under some suitable assumptions,we show that the sequence generated by the new algorithm is strong convergence.Moreover,the proposed algorithm needs only two projections onto a same half-space per iteration.Finally,inspired by Popov's method,we proposed a modified two-subgradient extragradient algorithm.The new algorithm needs only one value of underlying mapping and all the projections are computed by projecting a vector onto two different half-spaces per iteration,respectively.Under some mild assumptions,we show that the sequence generated by this algorithm is weak convergence.In the first chapter,we introduce the research background,the research status at home and abroad and the structure arrangement of this dissertation.In the second chapter,we introduce the definitions and lemmas needed to be used in the convergence analysis.In the third chapter,we propose a new modified subgradient extragradient algorithm.Under some mild assumptions,we prove the weak convergence of the algorithm.In the fourth chapter,we propose a new inertial two-subgradient extragradient algorithm.Under some suitable assumptions,we prove the strong convergence of the new algorithm.In the fifth chapter,inspired by Popov's method,we propose a modified two-subgradient extragradient algorithm.Under the same assumptions with the fourth chapter,we show that the newly proposed algorithm is weakly convergent.In the sixth chapter,we summarize and look forward to this dissertation.