Inertial Residual Projection Method (IRPM) for Approximating Solutions of Variational Inequality Problems

Abstract

This study proposes a new inertial residual projection method (IRPM) with or
without Halpern update for solving monotone variational inequality problems
(VIPs) in real Hilbert spaces. Existing explicit projection methods, including those
introduced by Noor et al., 2000a, 2000b, are limited by weak convergence
guarantees, multiple projection steps per iteration, and fixed step-size
dependence—factors that hinder their efficiency, robustness, and scalability. To
address these limitations, the proposed IRPM-H method integrates an inertial
extrapolation step for acceleration, Halpern-type anchoring for strong
convergence, and a residual-based adaptive step-size strategy that eliminates the
need for prior knowledge of Lipschitz constants. The algorithm is designed to solve
VIPs involving monotone operators such as linear mappings with positive
semidefinite matrices and gradients of convex functions. Under standard
monotonicity and continuity assumptions, we prove that the sequence generated
by the IRPM-H method converges strongly to a solution of the variational
inequality, which also satisfies the fixed-point formulation. Numerical illustrations
were given to justify the theretical assertons and to demonstrate the effectiveness
of the proposed models. The results shows that our model compete favourably with
other existing models cited in the literature.