Fej\'er monotone sequences revisited

TL;DR

This paper introduces a generalized concept of Fejér monotonicity, extending previous quantitative results and explaining convergence rates for algorithms like Dykstra's under metric regularity.

Contribution

It develops a localized and relativized generalization of Fejér monotonicity, broadening the applicability of convergence analysis for iterative algorithms.

Findings

Extension of quantitative convergence results to the new generalized framework

Application to Dykstra's algorithm demonstrating explicit convergence rates

Theoretical explanation for convergence under metric regularity assumptions

Abstract

In this paper we introduce a localized and relativized generalization of the usual concept of Fej\'er monotonicity together with uniform and quantitative versions thereof and show that the main quantitative results obtained by the 1st author together with Nicolae and Leu\c{s}tean in 2018 and with L\'opez-Acedo and Nicolae in 2019 respectively, extend to this generalization. Our framework, in particular, covers the sequence generated by the Dykstra algorithm while the latter is not Fej\'er-monotone in the ordinary sense. This gives a theoretical explanation why under a metric regularity assumption one obtains an explicit rate of convergence for Dykstra's algorithm which was proved recently by the 2nd author.