A relaxed-inertial proximal point algorithm for strongly quasiconvex equilibrium problems on Hadamard manifolds

Abstract

We study a proximal point type method for approximating solutions to equilibrium problems generated by pseudomonotone and strongly quasiconvex bifunctions over Hadamard manifolds, that is complete simply connected Riemannian manifolds of nonpositive sectional curvature. Next to the usual proximal step, the method we consider also incorporates an inertia step together with a subsequent over-relaxation, the latter of which is treated in the context of Hadamard manifolds, to our knowledge, for the first time. Making use of a quantitative approach towards such proximal methods for strongly quasiconvex optimization developed by the authors in previous work, we in particular provide effective arguments for the convergence of the method, yielding explicit, fast and very uniform rates of convergence for the distance of the iterates towards the solution. These results extend previous work by Grad, Lara and Marcavillaca on such a method over finite-dimensional Euclidean spaces for the first time to a nonlinear setting, with the quantitative estimates already being novel in the Euclidean case. In particular, our effective approach allows for a fine-grained view on the assumptions on the surrounding objects, so that we are able to either weaken or even fully discharge some previous assumptions.