Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization

Junfeng Yang*, Xiaoming YUAN

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

327 Citations (Scopus)


The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have closed-form solutions. In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived. Global convergence results of these linearized ALM and ADM are established under standard assumptions. Finally, we verify the effectiveness and efficiency of these new methods by some numerical experiments.

Original languageEnglish
Pages (from-to)301-329
Number of pages29
JournalMathematics of Computation
Issue number281
Publication statusPublished - 2013

Scopus Subject Areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Alternating direction method
  • Augmented Lagrangian method
  • Convex programming
  • Linearized
  • Low-rank
  • Nuclear norm


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