Advancing the Korpelevich method: a new approach to equilibrium problems on Hadamard manifolds

We devise a relaxed-inertial subgradient extragradient method for solving pseudomonotone equilibrium problems on Hadamard manifolds. The method uses a self-adaptive step size that does not depend on the Lipschitz constant. Under suitable conditions, we demonstrate that every sequence produced by the proposed algorithms converges to a solution of the equilibrium problem. We also prove linear convergence of the proposed method when the bifunction satisfies strong pseudomonotonicity and Lipschitz continuity. We then provide an application of our main result to variational inequality and optimization problems using the proposed algorithms. We conclude our paper with a numerical example to verify our convergence result.