Relaxed inertial subgradient extragradient methods for equilibrium problems in Hilbert spaces and their applications to image restoration

We introduce two extragradient methods that incorporate one-step inertial terms and self-adaptive step sizes for equilibrium problems in real Hilbert spaces. These methods synergistically combine inertial techniques and relaxation parameters to enhance convergence speed while ensuring superior performance in addressing pseudomonotone and Lipschitz continuous equilibrium problems. The first method is formulated to achieve weak convergence, whereas the second method guarantees strong convergence; both methods feature designed step-size adaptation mechanisms that maintain feasibility and efficiency. The proposed methods utilize adaptive step sizes that are updated at each iteration based on previous iterations. Convergence is demonstrated under mild assumptions, and our findings generalize and extend some related results within the existing literature. Lastly, we present numerical experiments that illustrate the performance of the proposed methods, including their applications to image restoration problems.