Moduli of regularity and rates of convergence for Fej\'er monotone sequences

TL;DR

This paper introduces the modulus of regularity to analyze convergence rates of Fejér monotone sequences, providing a unified framework for various regularity notions and algorithms in optimization and fixed point theory.

Contribution

It proposes the concept of modulus of regularity, unifying multiple regularity notions and deriving convergence rates for key algorithms in convex optimization.

Findings

Unified approach to convergence analysis using modulus of regularity

Quantitative rates for Picard, Mann, proximal point, and cyclic algorithms

A new quantitative version of the minimizers and zeros set equivalence

Abstract

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the finite termination, for classes of Fej\'er monotone sequences which appear in fixed point theory, monotone operator theory, and convex optimization. This concept allows for a unified approach to several notions such as weak sharp minima, error bounds, metric subregularity, H\"older regularity, etc., as well as to obtain rates of convergence for Picard iterates, the Mann algorithm, the proximal point algorithm and the cyclic algorithm. As a byproduct we obtain a quantitative version of the well-known fact that for a convex lower semi-continuous function the set of minimizers coincides with the set of zeros of its subdifferential and the set of fixed points of its resolvent.