Proximal point algorithm for convex minimisation on Banach spheres

Abstract

In this paper, we study a proximal point algorithm for convex minimisation on Banach spheres. In the spherical framework, we consider resolvent operators associated with proper lower semicontinuous spherically convex functions and investigate the asymptotic behaviour of iterative sequences generated by these resolvents. Under suitable geometric assumptions on the underlying Banach sphere and a sequential delta-continuity assumption on the duality mapping, we prove that the generated sequence delta-converges to a minimiser of the objective function. This result provides a delta-convergence principle for proximal point algorithms in the setting of Banach spheres.