Backward–forward algorithms for structured monotone inclusions in Hilbert spaces

Hédy Attouch, Juan Peypouquet*, Patrick Redont

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

18 Citations (Scopus)


In this paper, we study the backward–forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward–backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward–backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient-projection algorithms, and give a numerical illustration of theoretical interest.

Original languageEnglish
Pages (from-to)1095-1117
Number of pages23
JournalJournal of Mathematical Analysis and Applications
Issue number2
Publication statusPublished - 15-Jan-2018
Externally publishedYes


  • Forward–backward algorithm
  • Monotone inclusion
  • Proximal-gradient method

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