Effective strong convergence of the proximal point algorithm in CAT(0) spaces

TL;DR

This paper uses proof mining techniques to derive explicit convergence rates and metastability bounds for the proximal point algorithm in CAT(0) spaces, enhancing understanding of its convergence behavior.

Contribution

It introduces uniform quantitative bounds on the strong convergence of the proximal point algorithm in CAT(0) spaces, including rates of convergence and metastability for various conditions.

Findings

Computed convergence rates for uniformly convex functions

Derived metastability bounds in totally bounded CAT(0) spaces

Applied proof mining methods to analyze algorithm convergence

Abstract

We apply methods of proof mining to obtain uniform quantitative bounds on the strong convergence of the proximal point algorithm for finding minimizers of convex, lower semicontinuous proper functions in CAT(0) spaces. Thus, for uniformly convex functions we compute rates of convergence, while, for totally bounded CAT(0) spaces we apply methods introduced by Kohlenbach, the first author and Nicolae to compute rates of metastability.