On the convergence of splitting algorithm for mixed equilibrium problems on Hadamard manifolds

An explicit extragradient algorithm for equilibrium problems on Hadamard manifolds

“…4 The R-linear rate of the convergence Algorithms in [18] have some special advantages that they are done without the prior knowledge of the Lipschitz-type constants of the bifunction. However, in the case that bifunction f is strongly pseudomonotone (SP), the linear rate of convergence cannot be obtained for these algorithms.…”

“…Indeed, in recent years, various algorithms, which involves monotone bifunctions, have been extended to solve equilibrium problems from Hilbert spaces to the more general setting of Riemannian manifolds. In particular, Khammahawong et al [18] presented an extragradient algorithm to solve strongly pseudomonotone equilibrium problems on Hadamard manifolds. Their algorithm is described as follows.…”

“…The convergence of sequence {x n } was investigated and obtained. Inspired by the work in [10,18], the aim of this paper is to present an extragradient algorithm with new stepsize rules for pseudomonotone equilibrium problems on Hadamard manifolds and study its convergence properties. Our algorithm uses a variable stepsize sequence, which is generated at each iteration, based on some previous iterates, and without any linesearch procedure.…”

On the existence and approximation of solutions of generalized equilibrium problem on Hadamard manifolds

“…The advantages of doing this are well documented. For example, by choosing a suitable Riemannian metric, optimization problems with nonconvex objective functions can be viewed as convex [20,31,41]. Also, from the perspective of the Riemannian geometry, constrained optimization problems can be seen as unconstrained [31,32,41].…”

“…Furthermore, iterative approximation of the various optimization problems is another interesting area of research in nonlinear analysis, especially in the twin concepts of fixed point and optimization theory. For extensive literature on the iterative methods for solving variational inequalities (see [16,17,20,22,25,29] and the reference therein). Also, for methods of approximating a solution of the EP in both the linear and nonlinear spaces, see [1,20,28,40].…”