Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems

Howard Heaton, Yair Censor

Research output: Contribution to journalArticlepeer-review


The common fixed points problem requires finding a point in the intersection of fixed points sets of a finite collection of operators. Quickly solving problems of this sort is of great practical importance for engineering and scientific tasks (e.g., for computed tomography). Iterative methods for solving these problems often employ a Krasnosel’skiĭ–Mann type iteration. We present an asynchronous sequential inertial (ASI) algorithmic framework in a Hilbert space to solve common fixed points problems with a collection of nonexpansive operators. Our scheme allows use of out-of-date iterates when generating updates, thereby enabling processing nodes to work simultaneously and without synchronization. This method also includes inertial type extrapolation terms to increase the speed of convergence. In particular, we extend the application of the recent “ARock algorithm” (Peng et al. in SIAM J Sci Comput 38:A2851–A2879, 2016) in the context of convex feasibility problems. Convergence of the ASI algorithm is proven with no assumption on the distribution of delays, except that they be uniformly bounded. Discussion is provided along with a computational example showing the performance of the ASI algorithm applied in conjunction with a diagonally relaxed orthogonal projections (DROP) algorithm for estimating solutions to large linear systems.

Original languageEnglish
Pages (from-to)95-119
Number of pages25
JournalJournal of Global Optimization
Issue number1
StatePublished - 15 May 2019

Bibliographical note

Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.


  • Asynchronous sequential iterations
  • Convex feasibility problem
  • DROP algorithm
  • Fixed point iteration
  • Kaczmarz method
  • Nonexpansive operator

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics
  • Business, Management and Accounting (miscellaneous)
  • Computer Science Applications
  • Management Science and Operations Research


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