A relaxed averaging gradient projection method with linear convergence for quasi-variational inequalities

This paper addresses an open question in the analysis of quasi-variational inequalities (QVIs) by establishing the linear convergence of a relaxed averaging gradient projection method, which generalizes earlier schemes with restricted parameter choices. While recent results have achieved linear rates under strong monotonicity and Lipschitz continuity, they have been limited to specific relaxation regimes. We bridge this gap with a novel averaging-based algorithm that extends and unifies existing projection methods. Our main contributions are: (i) Linear Convergence Guarantee: We provide the proof of linear convergence for a relaxed averaging scheme applied to QVIs, valid for a broad range of averaging parameters that include both classical and non-standard choices. (ii) Generalized Framework: Our approach significantly expands the existing theory by introducing a flexible averaging step, which recovers known methods as special cases while enabling new algorithmic behaviours. This offers greater flexibility and potential for performance tuning. (iii) Computational Superiority: Extensive numerical experiments on benchmark problems confirm that our method consistently outperforms state-of-the-art alternatives in convergence speed and solution accuracy, especially in large-scale settings. This work offers a rigorous theoretical advancement and a more efficient computational tool for solving QVIs in Hilbert spaces.