Set-Valued and Variational Analysis

, Volume 18, Issue 3, pp 373–404

Dualization of Signal Recovery Problems

  • Patrick L. Combettes
  • Đinh Dũng
  • Bằng Công Vũ

DOI: 10.1007/s11228-010-0147-7

Cite this article as:
Combettes, P.L., Dũng, Đ. & Vũ, B.C. Set-Valued Anal (2010) 18: 373. doi:10.1007/s11228-010-0147-7


In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel–Moreau–Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.


Convex optimizationDenoisingDictionaryDykstra-like algorithmDualityForward-backward splittingImage reconstructionImage restorationInverse problemSignal recoveryPrimal-dual algorithmProximity operatorTotal variation

Mathematics Subject Classifications (2010)


Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Patrick L. Combettes
    • 1
  • Đinh Dũng
    • 2
  • Bằng Công Vũ
    • 1
  1. 1.Laboratoire Jacques-Louis Lions–UMR 7598UPMC Université Paris 06ParisFrance
  2. 2.Information Technology InstituteVietnam National UniversityHanoiVietnam