Three Adaptive Inertial Subgradient Exgradient Algorithms For Pseudomonotone Variational Inequalities

In optimization theory,variational inequality has always been a hot topic,and projection algorithm has been studied deeply because of its small amount of computation.Because the orthogonal projection of a projection on a non-empty closed convex set is not easy to calculate,Censor,Gibali and Reich proposed the subgradient external gradient method,and Duong Q L and Dang proposed the inertial subgradient external gradient method based on this method by integrating inertial technology,which greatly improves the convergence speed.However,the step size in this method is limited by the Lipschitz constant of the mapping and can only solve monotone variational inequalities.In view of the two shortcomings of this method,this paper proposes three adaptive inertial subgradient external gradient methods for solving pseudomonotone,combining with line search technology and modifying its search direction.Firstly,we propose a new inertial subgradient external gradient method with adaptive step size by combining the line search technique with the inertial subgradient external gradient method proposed by Duong Q L and Dang?Under the same assumption,when the mapping is pseudomonotone and continuous,the new method has weak convergence property,and the numerical experimental results also show that the performance of the new method is better.Secondly,considering the influence of the search direction on the convergence speed of the algorithm,two other methods are proposed based on the new algorithm by constantly improving the search direction,which also have the advantages of the new algorithm.The results of computer numerical experiments show that the numerical performance of the algorithm is improved with the continuous improvement of the search direction.It shows that it is meaningful to modify the search direction under the framework of this algorithm.