Proximal minimization in CAT$(\kappa)$ spaces

TL;DR

This paper establishes convergence results for proximal point algorithms in CAT(κ) spaces with positive curvature, extending existing methods to a new geometric setting using recent resolvent definitions.

Contribution

It introduces convergence analysis of proximal algorithms in CAT(κ) spaces with positive curvature, utilizing a novel resolvent concept by Kimura and Kohsaka.

Findings

Proximal point algorithm converges in CAT(κ) spaces with κ > 0.

Splitting variants also exhibit convergence under the new framework.

Extends convex optimization techniques to curved geometric spaces.

Abstract

In this note, we provide convergence results for the proximal point algorithm and a splitting variant thereof in the setting of CAT$(\kappa)$ spaces with $\kappa > 0$ using a recent definition for the resolvent of a convex, lower semi-continuous function due to Kimura and Kohsaka (J. Fixed Point Theory Appl. 18 (2016), 93-115).