Journal of Scientific Computing

, Volume 54, Issue 2–3, pp 428–453 | Cite as

Accelerated Linearized Bregman Method

  • Bo Huang
  • Shiqian Ma
  • Donald Goldfarb


In this paper, we propose and analyze an accelerated linearized Bregman (ALB) method for solving the basis pursuit and related sparse optimization problems. This accelerated algorithm is based on the fact that the linearized Bregman (LB) algorithm first proposed by Stanley Osher and his collaborators is equivalent to a gradient descent method applied to a certain dual formulation. We show that the LB method requires O(1/ϵ) iterations to obtain an ϵ-optimal solution and the ALB algorithm reduces this iteration complexity to \(O(1/\sqrt{\epsilon})\) while requiring almost the same computational effort on each iteration. Numerical results on compressed sensing and matrix completion problems are presented that demonstrate that the ALB method can be significantly faster than the LB method.


Convex optimization Linearized Bregman method Accelerated linearized Bregman method Compressed sensing Basis pursuit Matrix completion 



We would like to thank Wotao Yin for fruitful discussions and the anonymous referees for making several very helpful suggestions.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchColumbia UniversityNew YorkUSA
  2. 2.Institute for Mathematics and Its ApplicationsUniversity of MinnesotaMinneapolisUSA

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