An application of proof mining to the proximal point algorithm in CAT(0) spaces

TL;DR

This paper applies proof mining techniques to derive uniform rates of metastability for the proximal point algorithm in totally bounded CAT(0) spaces, providing quantitative insights into its convergence behavior.

Contribution

It introduces the first uniform rates of metastability for the proximal point algorithm in CAT(0) spaces using proof mining methods, especially in the totally bounded case.

Findings

Derived uniform rates of metastability for the algorithm

Extended proof mining techniques to CAT(0) spaces

Enhanced understanding of convergence in non-constructive proofs

Abstract

We compute, using techniques originally introduced by Kohlenbach, the first author and Nicolae, uniform rates of metastability for the proximal point algorithm in the context of CAT(0) spaces (as first considered by Bacak), specifically for the case where the ambient space is totally bounded. This result is part of the program of proof mining, which aims to apply methods of mathematical logic with the purpose of extracting quantitative information out of ordinary mathematical proofs, which may not be necessarily constructive.