An implementable proximal point algorithmic framework for nuclear norm minimization

Y.-J. Liu, Defeng Sun, K.-C. Toh

Research output: Journal article publicationJournal articleAcademic researchpeer-review

65 Citations (Scopus)


The nuclear norm minimization problem is to find a matrix with the minimum nuclear norm subject to linear and second order cone constraints. Such a problem often arises from the convex relaxation of a rank minimization problem with noisy data, and arises in many fields of engineering and science. In this paper, we study inexact proximal point algorithms in the primal, dual and primal-dual forms for solving the nuclear norm minimization with linear equality and second order cone constraints. We design efficient implementations of these algorithms and present comprehensive convergence results. In particular, we investigate the performance of our proposed algorithms in which the inner sub-problems are approximately solved by the gradient projection method or the accelerated proximal gradient method. Our numerical results for solving randomly generated matrix completion problems and real matrix completion problems show that our algorithms perform favorably in comparison to several recently proposed state-of-the-art algorithms. Interestingly, our proposed algorithms are connected with other algorithms that have been studied in the literature. © 2011 Springer and Mathematical Optimization Society.
Original languageEnglish
Pages (from-to)399-436
Number of pages38
JournalMathematical Programming
Issue number1-2
Publication statusPublished - 1 Jun 2012
Externally publishedYes


  • Accelerated proximal gradient method
  • Gradient projection method
  • Nuclear norm minimization
  • Proximal point method
  • Rank minimization

ASJC Scopus subject areas

  • Software
  • General Mathematics


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