Mathematical Programming Computation

, Volume 10, Issue 4, pp 703–743 | Cite as

QSDPNAL: a two-phase augmented Lagrangian method for convex quadratic semidefinite programming

  • Xudong LiEmail author
  • Defeng Sun
  • Kim-Chuan Toh
Full Length Paper


In this paper, we present a two-phase augmented Lagrangian method, called QSDPNAL, for solving convex quadratic semidefinite programming (QSDP) problems with constraints consisting of a large number of linear equality and inequality constraints, a simple convex polyhedral set constraint, and a positive semidefinite cone constraint. A first order algorithm which relies on the inexact Schur complement based decomposition technique is developed in QSDPNAL-Phase I with the aim of solving a QSDP problem to moderate accuracy or using it to generate a reasonably good initial point for the second phase. In QSDPNAL-Phase II, we design an augmented Lagrangian method (ALM) wherein the inner subproblem in each iteration is solved via inexact semismooth Newton based algorithms. Simple and implementable stopping criteria are designed for the ALM. Moreover, under mild conditions, we are able to establish the rate of convergence of the proposed algorithm and prove the R-(super)linear convergence of the KKT residual. In the implementation of QSDPNAL, we also develop efficient techniques for solving large scale linear systems of equations under certain subspace constraints. More specifically, simpler and yet better conditioned linear systems are carefully designed to replace the original linear systems and novel shadow sequences are constructed to alleviate the numerical difficulties brought about by the crucial subspace constraints. Extensive numerical results for various large scale QSDPs show that our two-phase algorithm is highly efficient and robust in obtaining accurate solutions. The software reviewed as part of this submission was given the DOI (Digital Object Identifier)


Quadratic semidefinite programming Schur complement Augmented Lagrangian Inexact semismooth Newton method 

Mathematics Subject Classification

90C06 90C20 90C22 90C25 65F10 


  1. 1.
    Alfakih, A.Y., Khandani, A., Wolkowicz, H.: Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12, 13–30 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. 86, 135–160 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Biswas, P., Liang, T.C., Toh, K.-C., Wang, T.C., Ye, Y.: Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Trans. Autom. Sci. Eng. 3, 360–371 (2006)CrossRefGoogle Scholar
  4. 4.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)CrossRefGoogle Scholar
  5. 5.
    Burkard, R.E., Karisch, S.E., Rendl, F.: QAPLIB—a quadratic assignment problem library. J. Global Optim. 10, 391–403 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, L., Sun, D.F., Toh, K.-C.: An efficient inexact symmetric Gauss–Seidel based majorized ADMM for high-dimensional convex composite conic programming. Math. Program. 161, 237–270 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cui, Y., Sun, D.F., Toh, K.-C.: On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions, arXiv:1610.00875 (2016)
  8. 8.
    Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  9. 9.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)zbMATHGoogle Scholar
  10. 10.
    Han, D., Sun, D., Zhang, L.: Linear rate convergence of the alternating direction method of multipliers for convex composite programming. Math. Oper. Res. (2017). MathSciNetCrossRefGoogle Scholar
  11. 11.
    Higham, N.J.: Computing the nearest correlation matrix—a problem from finance. IMA J. Numer. Anal. 22, 329–343 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jiang, K., Sun, D.F., Toh, K.-C.: An inexact accelerated proximal gradient method for large scale linearly constrained convex SDP. SIAM J. Optim. 22, 1042–1064 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jiang, K., Sun, D.F., Toh, K.-C.: A partial proximal point algorithm for nuclear norm regularized matrix least squares problems. Math. Program. Comput. 6, 281–325 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Krislock, N., Lang, J., Varah, J., Pai, D.K., Seidel, H.-P.: Local compliance estimation via positive semidefinite constrained least squares. IEEE Trans. Robot. 20, 1007–1011 (2004)CrossRefGoogle Scholar
  16. 16.
    Li, L., Toh, K.-C.: An inexact interior point method for l1-regularized sparse covariance selection. Math. Program. Comput. 2, 291–315 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li, X.D.: A Two-Phase Augmented Lagrangian Method for Convex Composite Quadratic Programming, PhD thesis, Department of Mathematics, National University of Singapore (2015)Google Scholar
  18. 18.
    Li, X.D., Sun, D.F., Toh, K.-C.: A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions. Math. Program. 155, 333–373 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nie, J.W., Yuan, Y.X.: A predictor-corrector algorithm for QSDP combining Dikin-type and Newton centering steps. Ann. Oper. Res. 103, 115–133 (2001)MathSciNetCrossRefGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Povh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6, 231–241 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Qi, H.D.: Local duality of nonlinear semidefinite programming. Math. Oper. Res. 34, 124–141 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Qi, H.D., Sun, D.F.: A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28, 360–385 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Conjugate Duality and Optimization, CBMS-NSF Regional Conf. Ser. Appl. Math. vol. 16. SIAM, Philadelphia (1974)Google Scholar
  25. 25.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sun, D.F., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150–169 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sun, D.F., Toh, K.-C., Yang, L.: A convergent 3-block semiproximal alternating direction method of multipliers for conic programming with 4-type constraints. SIAM J. Optim. 25, 882–915 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sun, D.F., Toh, K.-C., Yang, L.: An efficient inexact ABCD method for least squares semidefinite programming. SIAM J. Optim. 26, 1072–1100 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Sun, J., Zhang, S.: A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs. Eur. J. Oper. Res. 207, 1210–1220 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Toh, K.-C.: An inexact primal-dual path following algorithm for convex quadratic SDP. Math. Program. 112, 221–254 (2008)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Toh, K.-C., Tütüncü, R., Todd, M.: Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems. Pac. J. Optim. 3, 135–164 (2007)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Yang, L., Sun, D.F., Toh, K.-C.: SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints. Math. Program. Comput. 7, 331–366 (2015)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhao, X.Y.: A Semismooth Newton-CG Augmented Lagrangian Method for Large Scale Linear and Convex Quadratic SDPs, PhD thesis, Department of Mathematics, National University of Singapore (2009)Google Scholar
  35. 35.
    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)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  1. 1.Department of Operations Research and Financial EngineeringPrinceton University, Sherrerd HallPrincetonUSA
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung HomHong Kong
  3. 3.Department of Mathematics, and Institute of Operations Research and AnalyticsNational University of SingaporeSingaporeSingapore

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