An abstract proximal point algorithm

TL;DR

This paper introduces an abstract proximal point algorithm framework that generalizes convergence proofs for convex optimization problems in CAT(0) and Hilbert spaces, utilizing nonexpansive mappings and proof mining techniques.

Contribution

It abstracts the resolvent concept to a broader class of mappings, providing unified weak and strong convergence proofs with computable rates in non-linear spaces.

Findings

Weak convergence in CAT(0) spaces established

Strong convergence under uniform firm nonexpansivity shown

Computable convergence rates derived using proof mining

Abstract

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.