On efficiently solving the subproblems of a level-set method for fused lasso problems

Xudong Li, Defeng Sun, Kim Chuan Toh

Research output: Journal article publicationJournal articleAcademic researchpeer-review

17 Citations (Scopus)


In applying the level-set method developed in [E. Van den Berg and M. P. Friedlander, SIAM J. Sci. Comput., 31 (2008), pp. 890–912] and [E. Van den Berg and M. P. Friedlander, SIAM J. Optim., 21 (2011), pp. 1201–1229] to solve the fused lasso problems, one needs to solve a sequence of regularized least squares subproblems. In order to make the level-set method practical, we develop a highly efficient inexact semismooth Newton based augmented Lagrangian method for solving these subproblems. The efficiency of our approach is based on several ingredients that constitute the main contributions of this paper. First, an explicit formula for constructing the generalized Jacobian of the proximal mapping of the fused lasso regularizer is derived. Second, the special structure of the generalized Jacobian is carefully extracted and analyzed for the efficient implementation of the semismooth Newton method. Finally, numerical results, including the comparison between our approach and several state-of-the-art solvers, on real data sets are presented to demonstrate the high efficiency and robustness of our proposed algorithm in solving challenging large-scale fused lasso problems.

Original languageEnglish
Pages (from-to)1842-1866
Number of pages25
JournalSIAM Journal on Optimization
Issue number2
Publication statusPublished - 20 Jun 2018


  • Convex composite programming
  • Fused lasso
  • Generalized Jacobian
  • Level-set method
  • Semismooth Newton method

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

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