Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists

Laurent Pierre Condat, Daichi Kitahara, Andrés Contreras, Akira Hirabayashi

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview, we demystify a selection of recent proximal splitting algorithms: we present them within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metrics. Along the way, we easily derive new variants of the algorithms and revisit existing convergence results, extending the parameter ranges in several cases. In particular, we emphasize that when the smooth term in the objective function is quadratic, e.g., for least-squares problems, convergence is guaranteed with larger values of the relaxation parameter than previously known. Such larger values are usually beneficial for the convergence speed in practice.
Original languageEnglish (US)
Pages (from-to)375-435
Number of pages61
JournalSIAM Review
Issue number2
StatePublished - May 9 2023

Bibliographical note

KAUST Repository Item: Exported on 2023-07-17
Acknowledgements: The first author did part of this work during a stay at Ritsumeikan University in 2018, hosted by the second and fourth authors, thanks to a fellowship from the Japanese Society for the Promotion of Science (JSPS), L17565. The third author contributed to this work during his visit to the first author at GIPSA-lab, Grenoble, France, in 2019. He was supported by the CMM-ANID PIA grant AFB170001.

ASJC Scopus subject areas

  • Computational Mathematics
  • Theoretical Computer Science
  • Applied Mathematics


Dive into the research topics of 'Proximal Splitting Algorithms for Convex Optimization: A Tour of Recent Advances, with New Twists'. Together they form a unique fingerprint.

Cite this