Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective

Bingsheng He, Xiaoming YUAN*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

180 Citations (Scopus)

Abstract

Recently, some primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation image restoration. This paper focuses on the convergence analysis of these primal-dual algorithms and shows that their involved parameters (including step sizes) can be significantly enlarged if some simple correction steps are supplemented. Some new primal-dual-based methods are thus proposed for solving the saddle-point problem. We show that these new methods are of the contraction type: the iterative sequences generated by these new methods are contractive with respect to the solution set of the saddle-point problem. The global convergence of these new methods thus can be obtained within the analytic framework of contraction-type methods. The novel study on these primal-dual algorithms from the perspective of contraction methods substantially simplifies existing convergence analysis. Finally, we show the efficiency of the new methods numerically.

Original languageEnglish
Pages (from-to)119-149
Number of pages31
JournalSIAM Journal on Imaging Sciences
Volume5
Issue number1
DOIs
Publication statusPublished - 2012

Scopus Subject Areas

  • Mathematics(all)
  • Applied Mathematics

User-Defined Keywords

  • Contraction method
  • Image restoration
  • Primal-dual method
  • Proximal point algorithm
  • Saddle point problem
  • Total variation

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