Proximal Point Algorithms with Inertial Extrapolation for Quasi-convex Pseudo-monotone Equilibrium Problems

In this paper, we study the proximal point algorithm with inertial extrapolation to approximate a solution to the quasi-convex pseudo-monotone equilibrium problem. In the proposed algorithm, the inertial parameter is allowed to take both negative and positive values during implementations. The possibility of the choice of negative values for the inertial parameter sheds more light on the range of values of the inertial parameter for the proximal point algorithm. Under standard assumptions, we prove that the sequence of iterates generated by the proposed algorithm converges to a solution of the equilibrium problem when the bifunction is strongly quasi-convex in its second argument. Sublinear and linear rates of convergence are also given under standard conditions. Numerical results are reported for both cases of negative and positive inertial factor of the proposed algorithm and comparison with related algorithm is discussed.