The extragradient algorithm with inertial effects extended to equilibrium problems

Abstract

In this paper, two algorithms are proposed for a class of pseudomonotone and strongly pseudomonotone equilibrium problems. These algorithms can be viewed as a extension of the paper title, the extragradient algorithm with inertial effects for solving the variational inequality proposed by Dong et al. (Optimization 65:2217-2226, 2016. 10.1080/02331934.2016.1239266). The weak convergence of the first algorithm is well established based on the standard assumption imposed on the cost bifunction. We provide a strong convergence for the second algorithm without knowing the strongly pseudomonoton and the Lipschitz-type constants of cost bifunction. The practical interpretation of a second algorithm is that the algorithm uses a sequence of step sizes that is converging to zero and non-summable. Numerical examples are used to assist the well-established convergence result, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.