Approximate subgradient extragradient methods for solving variational inequality problems: Convergence analysis and applications in signal and image processing

This paper introduces two new inertial algorithms for solving variational inequality problems in a real Hilbert space. The first algorithm integrates an acceleration scheme into a two-step subgradient extragradient framework, guaranteeing weak convergence under monotonicity and achieving R -linear convergence when the operator is strongly monotone. The second algorithm modifies this structure to ensure strong convergence of the generated sequence. Both methods employ adaptive step size strategies, which eliminate the need for estimating the Lipschitz constant or performing line search procedures. The effectiveness of the proposed algorithms is validated through numerical experiments, which demonstrate their competitive performance relative to existing methods. Their practical applicability is further illustrated through applications in signal recovery and image restoration, where they outperform several well-known algorithms from the literature.