An Adaptive Two-Stage Proximal Algorithm for Equilibrium Problems in Hadamard Spaces

Equilibrium problems in Hadamard metric spaces are considered in the paper. For approximate solution of problems, a new iterative adaptive two-stage proximal algorithm is proposed and analyzed. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of the value of bifunction’s Lipschitz constants. For pseudo-monotone bifunctions of Lipschitz type, the theorem on weak convergence of the sequences generated by the algorithm is proved. It is shown that the proposed algorithm is applicable to pseudo-monotone variational inequalities in Hilbert spaces.