Mathematical Programming Computation

, Volume 7, Issue 3, pp 331–366 | Cite as

SDPNAL\(+\): a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints

  • Liuqin Yang
  • Defeng Sun
  • Kim-Chuan TohEmail author
Full Length Paper


In this paper, we present a majorized semismooth Newton-CG augmented Lagrangian method, called SDPNAL\(+\), for semidefinite programming (SDP) with partial or full nonnegative constraints on the matrix variable. SDPNAL\(+\) is a much enhanced version of SDPNAL introduced by Zhao et al. (SIAM J Optim 20:1737–1765, 2010) for solving generic SDPs. SDPNAL works very efficiently for nondegenerate SDPs but may encounter numerical difficulty for degenerate ones. Here we tackle this numerical difficulty by employing a majorized semismooth Newton-CG augmented Lagrangian method coupled with a convergent 3-block alternating direction method of multipliers introduced recently by Sun et al. (SIAM J Optim, to appear). Numerical results for various large scale SDPs with or without nonnegative constraints show that the proposed method is not only fast but also robust in obtaining accurate solutions. It outperforms, by a significant margin, two other competitive publicly available first order methods based codes: (1) an alternating direction method of multipliers based solver called SDPAD by Wen et al. (Math Program Comput 2:203–230, 2010) and (2) a two-easy-block-decomposition hybrid proximal extragradient method called 2EBD-HPE by Monteiro et al. (Math Program Comput 1–48, 2014). In contrast to these two codes, we are able to solve all the 95 difficult SDP problems arising from the relaxations of quadratic assignment problems tested in SDPNAL to an accuracy of \(10^{-6}\) efficiently, while SDPAD and 2EBD-HPE successfully solve 30 and 16 problems, respectively. In addition, SDPNAL\(+\) appears to be the only viable method currently available to solve large scale SDPs arising from rank-1 tensor approximation problems constructed by Nie and Wang (SIAM J Matrix Anal Appl 35:1155–1179, 2014). The largest rank-1 tensor approximation problem we solved (in about 14.5 h) is nonsym(21,4), in which its resulting SDP problem has matrix dimension \(n = 9261\) and the number of equality constraints \(m =12{,}326{,}390\).


Semidefinite programming Degeneracy Augmented Lagrangian Semismooth Newton-CG method 

Mathematics Subject Classification

90C06 90C22 90C25 65F10 



The authors would like to thank Jiawang Nie and Li Wang for sharing their codes on semidefinite relaxations of rank-1 tensor approximation problems.


  1. 1.
    Arnold, V.I.: On matrices depending on parameters. Russ. Math. Surv. 26, 29–43 (1971)CrossRefGoogle Scholar
  2. 2.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization: Theory and Examples, vol. 3. Springer, Berlin (2006)Google Scholar
  4. 4.
    Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120, 479–495 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burer, S., Monteiro, R.D., Zhang, Y.: A computational study of a gradient-based log-barrier algorithm for a class of large-scale sdps. Math. Program. 95, 359–379 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, C., He, B., Ye, Y., Yuan, X.: The direct extension of admm for multi-block convex minimization problems is not necessarily convergent. Math. Program. (to appear)Google Scholar
  7. 7.
    Eisenblätter, A., Grötschel, M., Koster, A.M.: Frequency planning and ramifications of coloring. Discuss. Math. Graph Theory 22, 51–88 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hahn, P., Anjos, M.: QAPLIB—a quadratic assignment problem library.
  9. 9.
    Hiriart-Urruty, J.-B., Strodiot, J.-J., Nguyen, V.H.: Generalized Hessian matrix and second-order optimality conditions for problems with \(C^{1,1}\) data. Appl. Math. Optim. 11, 43–56 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Monteiro, R., Ortiz, C., Svaiter, B.: A first-order block-decomposition method for solving two-easy-block structured semidefinite programs. Math. Program. Comput. 6, 103–150 (2014)Google Scholar
  11. 11.
    Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cones mutuellement polaires. C. R. Acad. Sci. 255, 238–240 (1962)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Nie, J., Wang, L.: Regularization methods for SDP relaxations in large-scale polynomial optimization. SIAM J. Optim. 22, 408–428 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nie, J., Wang, L.: Semidefinite relaxations for best rank-1 tensor approximations. SIAM J. Matrix Anal. Appl. 35, 1155–1179 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pang, J.-S., Sun, D.-F., Sun, J.: Semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems. Math. Oper. Res. 28, 39–63 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Peng, J., Wei, Y.: Approximating k-means-type clustering via semidefinite programming. SIAM J. Optim. 18, 186–205 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discret. Optim. 6, 231–241 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Rockafellar, R.T.: Conjugate duality and optimization. CBMS-NSF Regional Conf. Ser. Appl. Math. vol. 16. SIAM, Philadelphia (1974)Google Scholar
  18. 18.
    Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sloane, N.: Challenge problems: independent sets in graphs. (2005)
  21. 21.
    Sun, D.-F., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150–169 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sun, D.-F., Toh, K.-C., Yang, L.-Q.: A convergent 3-block semi-proximal alternating direction method of multipliers for conic programming with 4-type constraints. SIAM J. Optim. (to appear)Google Scholar
  23. 23.
    Toh, K.C.: Solving large scale semidefinite programs via an iterative solver on the augmented systems. SIAM J. Optim. 14, 670–698 (2004)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Trick, M., Chvatal, V., Cook, B., Johnson, D., McGeoch, C., Tarjan, R.: The second dimacs implementation challenge—NP hard problems: maximum clique, graph coloring, and satisfiability. (1992)
  25. 25.
    Wen, Z., Goldfarb, D., Yin, W.: Alternating direction augmented Lagrangian methods for semidefinite programming. Math. Program. Comput. 2, 203–230 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhao, X.-Y., Sun, D.-F., Toh, K.-C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and The Mathematical Programming Society 2015

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Department of Mathematics and Risk Management InstituteNational University of SingaporeSingaporeSingapore

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