Equilibrium Programming and New Iterative Methods in Hilbert Spaces

In this paper, we introduce a new iterative procedure for approximating a solution of an equilibrium problem involving a monotone and Lipschitz-type bifunction in a Hilbert space. The method includes two computational steps of proximal-like mapping incorporated with regularization terms. Several simple stepsize rules without linesearch are studied which allows the method to be more easily implemented with or without the information on the Lipschitz-type constant of cost bifunction. When the regularization parameter is suitably chosen, the iterative sequences generated by the method converge strongly to a particular solution of the problem. It turns out that the obtained solution by the method is the solution of a bilevel equilibrium problem whose constraint is the solution set of the considered equilibrium problem. Some numerical examples are computed to illustrate the computational effectiveness of the new method, and also to compare it with existing ones.