A Hybrid Projection Extragradient Method for Variational Inequality and Hierarchical Fixed-Point Problems

This study proposes a new strongly convergent iterative framework obtained by combining a Krasnosel’skiǐ–Mann type subgradient extragradient process with a hybrid projection strategy and an inertial extrapolation mechanism. The method is applied to address hierarchical fixed-point problems (HFPPs) for nonexpansive and quasi-nonexpansive mappings as well as variational inequality problems (VIPs) involving a pseudomonotone operator in real Hilbert spaces. The proposed scheme employs step sizes that are restricted by the inverse of the Lipschitz constant of the underlying cost operator. Strong convergence of the iterates is achieved under mild hypotheses on the inertial parameter and control sequences. The method is further applied to problems arising in optimization and monotone operator theory. The results show that the proposed framework generalizes and integrates a number of existing approaches while offering improved computational performance.