An Improved Inertial Viscosity Tseng-Type Method for Pseudomonotone variational Inequalities in Hilbert Spaces

In this paper, we propose a single inertial viscosity Tseng-type algorithm for solving the variational inequality and fixed-point problem in real Hilbert spaces. The method integrates inertial extrapolation, viscosity approximation, and a Mann-type iterative scheme with an adaptive step-size strategy. The cost operator is assumed to be pseudomonotone and Lipschitz continuous, while the fixed-point mapping is quasi-nonexpansive and demiclosed at zero. Unlike earlier approaches that require prior knowledge of the Lipschitz constant or impose stronger contractive conditions, the proposed algorithm employs an adaptive step-size rule that eliminates the need to estimate this constant in advance. Under appropriate assumptions, we establish the strong convergence of the generated sequence to a unique solution in the intersection of the variational inequality solution set and the fixed-point set. Numerical experiments comparing the proposed method with other existing methods demonstrate improved convergence speed, enhanced robustness with respect to parameter selection, and better computational efficiency, confirming the practical effectiveness of the algorithm.