Convergence Theorem of Inertial Projection and Contraction Methods for Pseudomonotone Variational Inequalities Problem

Abstract

This paper presents a novel convergence theorem for an inertial projection and contraction-based method aimed at solving pseudomonotone variational inequality problems (VIP) in real Hilbert spaces. The proposed algorithm combines inertial dynamics and contraction projections to enhance the convergence speed and stability of solutions, even when the operator involved is not strongly monotone but pseudomonotone. We establish the strong convergence of the method under mild conditions, demonstrating its effectiveness in non-strongly monotone settings. The theoretical analysis is complemented by numerical experiments, which show that the proposed method significantly outperforms classical projection and extragradient methods in terms of both computational efficiency and iteration count. This work contributes to the broader field of nonlinear optimization by providing a robust and efficient solution approach for VIP with pseudomonotone operators, and it opens pathways for further research on adaptive and large-scale extensions of the method.