Non-monotonic and self-adaptive strongly convergent iterative methods for efficiently solving variational inequalities with pseudomonotone operators

In this paper, we study two methods exhibiting strong convergence for solving classical variational inequality problems with Lipschitz-continuous and pseudomonotone operators in a real Hilbert space. These methods are inspired by Tseng's extragradient method, as well as the viscosity and Mann-type methods, both incorporating a straightforward step-size rule. These methods use variable step sizes that are modified at each iteration and dependent on prior iterations. These methods also have the additional advantage of not requiring prior knowledge of the Lipschitz constant or a linesearch process. The convergence for both methods is established under mild conditions. A series of numerical experiments is performed to validate the efficacy and superiority of the presented iterative methods over existing state-of-the-art methods. These experiments serve to affirm the effectiveness of the proposed methods in solving variational inequality problems in Hilbert spaces.