A dual spectral projected gradient method for log-determinant semidefinite problems

Takashi Nakagaki, Mituhiro Fukuda, Sunyoung Kim, Makoto Yamashita

Research output: Contribution to journalArticlepeer-review


We extend the result on the spectral projected gradient method by Birgin et al. in 2000 to a log-determinant semidefinite problem with linear constraints and propose a spectral projected gradient method for the dual problem. Our method is based on alternate projections on the intersection of two convex sets, which first projects onto the box constraints and then onto a set defined by a linear matrix inequality. By exploiting structures of the two projections, we show that the same convergence properties can be obtained for the proposed method as Birgin’s method where the exact orthogonal projection onto the intersection of two convex sets is performed. Using the convergence properties, we prove that the proposed algorithm attains the optimal value or terminates in a finite number of iterations. The efficiency of the proposed method is illustrated with the numerical results on randomly generated synthetic/deterministic data and gene expression data, in comparison with other methods including the inexact primal–dual path-following interior-point method, the Newton-CG primal proximal-point algorithm, the adaptive spectral projected gradient method, and the adaptive Nesterov’s smooth method. For the gene expression data, our results are compared with the quadratic approximation for sparse inverse covariance estimation method. We show that our method outperforms the other methods in obtaining a better objective value fast.

Original languageEnglish
Pages (from-to)33-68
Number of pages36
JournalComputational Optimization and Applications
Issue number1
StatePublished - 1 May 2020

Bibliographical note

Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.


  • Computational efficiency
  • Dual problem
  • Dual spectral projected gradient methods
  • Log-determinant semidefinite programs with linear constraints
  • Theoretical convergence results


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