Weak and Strong Convergence of Generalized Proximal Point Algorithms with Relaxed Parameters

TL;DR

This paper develops a comprehensive framework for generalized proximal point algorithms with relaxed parameters, establishing conditions for their convergence and linking iteration boundedness to the solution set of maximally monotone operators.

Contribution

It introduces new conditions on regularization and relaxation parameters that ensure convergence and equivalence between boundedness and solution existence, improving existing results.

Findings

Established conditions for weak and strong convergence.

Linked boundedness of iterations to the non-emptiness of the solution set.

Provided comparisons showing improvements over prior results.

Abstract

In this work, we propose and study a framework of generalized proximal point algorithms associated with a maximally monotone operator. We indicate sufficient conditions on the regularization and relaxation parameters of generalized proximal point algorithms for the equivalence of the boundedness of the sequence of iterations generated by this algorithm and the non-emptiness of the zero set of the maximally monotone operator, and for the weak and strong convergence of the algorithm. Our results cover or improve many results on generalized proximal point algorithms in our references. Improvements of our results are illustrated by comparing our results with related known ones.