Modified Subgradient Extragraduent Algorithm For Solving Monotone Variational Inequality Problems

The problem of variational inequality is a very important research topic in the optimization theory,and many experts and scholars have studied it.Among the algorithms to solve the variational inequality,the projection algorithm has been concerned by scholars because of its advantages such as small amount of calculation.In the projection algorithm,the orthogonal projection onto the non-empty closed convex set is not easy to calculate,so in order to solve this problem,Censor proposed the subgradient extragradient algorithm.Duong and Dang improved the subgradient extragradient algorithm proposed by Censor and proposed the inertial subgradient extragradient algorithm,which accelerated the convergence speed of the algorithm to a certain extent,but only proved the weak convergence of the algorithm.Therefore,it is of certain research value to construct the algorithm with strong convergence.Based on the algorithm proposed by Duong and Dang,two algorithms with strong convergence are proposed by combining viscosity approximation method.Firstly,we propose an inertial subgradient extragradient algorithm based on the viscosity approximation method for Duong and Dang's inertial subgradient extragradient algorithm,and prove the strong convergence of the algorithm under the condition that the mapping is monotone and Lipschitz continuous.Secondly,because the viscosity inertial subgradient extragradient algorithm still needs to calculate a projection onto the non-empty closed convex set,we extend the modified subgradient extragradient algorithm proposed by Yang Yantao.In order to accelerate the convergence speed of the algorithm,we joined the inertial thought,namely combines the inertial subgradient extragradient algorithm proposed by Duong and Dang.And joined the viscosity approximation method,in order to enhance the convergence of the algorithm.We construct a viscosity inertial double subgradient algorithm for solving the public yuan of the solution set of a class of monotone variational inequality problems in the Hilbert space and the fixed point set of the type of the nonexpanding mappings,and prove the strong convergence of the algorithm.