Linear Convergence of Subgradient Algorithm for Convex Feasibility on Riemannian Manifolds

We study the convergence issue of the subgradient algorithm for solving the convex feasibility problems in Riemannian manifolds, which was first proposed and analyzed by Bento and Melo [J. Optim. Theory Appl., 152 (2012), pp. 773--785]. The linear convergence property about the subgradient algorithm for solving the convex feasibility problems with the Slater condition in Riemannian manifolds are established, and some step sizes rules are suggested for finite convergence purposes, which are motivated by the work due to De Pierro Iusem [Appl. Math. Optim., 17 (1988), pp. 225--235]. As a by-product, the convergence result of this algorithm is obtained for the convex feasibility problem without the Slater condition assumption. These results extend and/or improve the corresponding known ones in both the Euclidean space and Riemannian manifolds.