Fejér monotone sequences revisited

Abstract

In this paper we introduce a localized and relativized generalization of the usual concept of Fejér monotonicity together with uniform and quantitative versions thereof and show that the main quantitative results obtained by the 1st author together with Nicolae and Leuştean in 2018 and with López-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 Fejér-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.

13pp