Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems

In this paper, basing on the subgradient extragradient method and inertial method with line-search process, we introduce two new algorithms for finding a common element of the solution set of a variational inequality and the fixed point set of a quasi-nonexpansive mapping with a demiclosedness property. The weak convergence of the algorithms are established under standard assumptions imposed on cost operators. The proposed algorithms can be considered as an improvement of the previously known inertial extragradient method over each computational step. Finally, for supporting the convergence of the proposed algorithms, we also consider several preliminary numerical experiments on a test problem.