The proximal point algorithm in geodesic spaces with curvature bounded above

TL;DR

This paper studies the convergence properties of the proximal point algorithm in geodesic spaces with curvature bounds, establishing conditions for the existence and convergence to minimizers of convex functions.

Contribution

It introduces the notion of resolvents in curved spaces and proves convergence results for the proximal point algorithm in these settings.

Findings

Sequences generated by the algorithm converge to minimizers.

Existence of minimizers is guaranteed under boundedness assumptions.

Convergence is established in complete 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.