The proximal point algorithm in geodesic spaces with curvature bounded above

Abstract

We investigate the asymptotic behavior of sequences generated by the proximal point algorithm for convex functions in complete geodesic spaces with curvature bounded above. Using the notion of resolvents of such functions, which was recently introduced by the authors, we show the existence of minimizers of convex functions under the boundedness assumptions on such sequences as well as the convergence of such sequences to minimizers of given functions.

This manuscript was accepted for publication in the journal Linear and Nonlinear Analysis (Submitted on 4 August 2016; Revised on 29 September 2016). The work in this manuscript had been done earlier than that in arXiv:1704.01360. Some results in this manuscript were generalized in arXiv:1704.01360