Golden Ratio‐Inspired Subgradient Extragradient Algorithms With Increasing Self‐Adaptive Step Sizes for Solving Equilibrium Problems and Applications to Image Restoration

ABSTRACT Equilibrium problems (EPs) provide a unified mathematical framework encompassing a broad class of models in optimization, variational inequalities, game theory, and applied sciences. In this paper, we propose two novel subgradient extragradient algorithms inspired by the golden ratio technique (GRT) for solving EPs in real Hilbert spaces. Both algorithms employ computationally efficient projections onto suitably constructed half‐spaces rather than full projections onto the feasible set, thereby reducing the per‐iteration computational cost. A key feature of our schemes is a self‐adaptive step‐size rule with increasing behavior, which updates the step sizes dynamically without requiring any prior knowledge of Lipschitz‐type constants. The first algorithm integrates golden‐ratio‐based extrapolation with subgradient projection steps, while the second incorporates an alternating extrapolation mechanism to further enhance numerical stability and efficiency. Under standard assumptions, we establish weak convergence of the generated sequences to a solution of the EP, and we additionally prove ‐linear convergence under stronger conditions. Extensive numerical experiments, including applications to image restoration, confirm that the proposed methods consistently outperform several existing extragradient‐type algorithms in terms of convergence speed, accuracy, and stability.