# Metastability of the proximal point algorithm with multi-parameters

## TL;DR

This paper applies proof mining techniques to analyze the convergence behavior of a generalized multi-parameter proximal point algorithm, providing explicit bounds and quantitative insights into its metastability and regularity.

## Contribution

It offers the first primitive recursive bounds on the metastability of a multi-parameter proximal point algorithm using proof mining methods.

## Key findings

- Provides explicit bounds on the metastability of the algorithm.
- Quantifies the asymptotic regularity of the iteration.
- Arithmetizes the $\

## Abstract

In this article we use techniques of proof mining to analyse a result, due to Yonghong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor's result ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided by Fernando Ferreira and Paulo Oliva's bounded functional interpretation, provides a primitive recursive bound on the metastability for the convergence of the algorithm, in the sense of Terence Tao. Furthermore, we obtain quantitative information on the asymptotic regularity of the iteration. The results of this paper are made possible by an arithmetization of the $\limsup$.