On the Convergence of the Inexact Running Krasnosel'skii-Mann Method

Andrey Bernstein, Emiliano Dall'Anese, Andrea Simonetto

Research output: Contribution to journalArticlepeer-review

2 Scopus Citations

Abstract

This letter leverages a framework based on averaged operators to tackle the problem of tracking fixed points associated with maps that evolve over time. In particular, this letter considers the Krasnosel'ski-Mann (KM) method in a settings where: 1) the underlying map may change at each step of the algorithm, thus leading to a 'running' implementation of the KM method, and 2) an imperfect information of the map may be available. An imperfect knowledge of the maps can capture cases where processors feature a finite precision or quantization errors, or the case where (part of) the map is obtained from measurements. The analytical results are applicable to inexact running algorithms for solving optimization problems, whenever the algorithmic steps can be written in the form of (a composition of) averaged operators; examples are provided for inexact running gradient methods and the forward-backward splitting method. Convergence of the average fixed-point residual is investigated for the non-expansive case; linear convergence to a unique fixed-point trajectory is showen in the case of inexact running algorithms emerging from contractive operators.

Original languageAmerican English
Article number8703134
Pages (from-to)613-618
Number of pages6
JournalIEEE Control Systems Letters
Volume3
Issue number3
DOIs
StatePublished - Jul 2019

Bibliographical note

Publisher Copyright:
© 2017 IEEE.

NREL Publication Number

  • NREL/JA-5D00-73860

Keywords

  • Numerical algorithms
  • optimization

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