Splitting Proximal Algorithms for Convex Optimizations over Metric Spaces with Curvature Bounded Above

Sakan Termkaew, Poom Kumam, Parin Chaipunya

Abstract

In this paper, we consider a splitting method combined with proximal methods for minimizing the sum of convex functions, where the proximal operators are defined by the curvature-adapted regularizations. Presented in the paper are two convergence theorems showing strong convergence under the assumptions of either the ambient space is locally compact or the objective function is uniformly convex with some specific modulations. We also present applications to solve convex feasibility problems, centroid problems, and particularly the Karcher means, where our main results can be useful. Finally, we include a series of numerical implementations of our algorithms to approximate the Karcher means of some randomly generated datasets fitted on the Lobachevskii plane.