The Glowinski-Le Tallec Splitting Method Revisited: A General Convergence And Convergence Rate Analysis

Yaonan Ma*, Lizhi LIAO

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we focus on a splitting method called the θ-scheme proposed by Glowinski and Le Tallec in [17, 20, 27]. First, we present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact θ-scheme. Second, for unconstrained problems, we prove the convergence of the θ-scheme and show a sublinear convergence rate in terms of objective value. Furthermore, a practical inexact θ-scheme is derived to solve l2-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the θ-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the θ-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the θ-scheme and our proposed methods.

Original languageEnglish
Pages (from-to)1681-1711
Number of pages31
JournalJournal of Industrial and Management Optimization
Volume17
Issue number4
DOIs
Publication statusPublished - Jul 2021

Scopus Subject Areas

  • Business and International Management
  • Strategy and Management
  • Control and Optimization
  • Applied Mathematics

User-Defined Keywords

  • convergence
  • convergence rate
  • inexactness criterion
  • Splitting method
  • the θ-scheme

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