Convergence of Adaptive Algorithms for Equilibrium Problems in Hadamard Spaces

In this paper, we consider the equilibrium problems under the setting of Hadamard spaces. For an approximate solution of equilibrium problems, iterative adaptive two-stage proximal algorithms are proposed and studied. At each step of the algorithms, the sequential minimization of two special strongly convex functions should be done. Our self-adaptive algorithms do not calculate bifunction values at additional points and do not require knowledge of the bifunction’s Lipschitz constants. For pseudomonotone bifunctions of the Lipschitz type that are weakly upper semicontinuous in the first variable and convex and lower semicontinuous in the second variable, we prove convergence theorems about sequences generated by proposed algorithms. First, we show weak convergence of generated sequences to a solution of the equilibrium problem. Then we prove strong convergence of Halpern regularization of adaptive extraproximal algorithm. Also it is shown that proposed algorithms are applicable to variational inequalities with Lipschitz-continuous, sequentially weakly continuous and pseudo-monotone operators acting in Hilbert spaces.