Abstract
This paper presents a Newton-CG augmented Lagrangian method for solving convex quadratically constrained quadratic semidefinite programming (QCQSDP) problems. Based on the Robinson’s CQ, the strong second order sufficient condition, and the constraint nondegeneracy conditions, we analyze the global convergence of the proposed method. For the inner problems, we prove the equivalence between the positive definiteness of the generalized Hessian of the objective functions in those inner problems and the constraint nondegeneracy of the corresponding dual problems, which guarantees the superlinear convergence of the inexact semismooth Newton-CG method to solve the inner problem. Numerical experiments show that the proposed method is very efficient to solve the large-scale convex QCQSDP problems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Sun, J., Zhang, S.: A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs. Eur. J. Oper. Res. 207, 1210–1220 (2010)
Zhao, X.Y.: A semismooth Newton-CG augmented Lagrangian method for large scale linear and convex quadratic semidefinite programming. Ph.D. thesis, National University of Singapore (2009)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1, 97–116 (1976)
Bonnans, J.F., Shapriro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Sun, D.F.: The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31, 761–776 (2006)
Sun, D.F., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150–169 (2002)
Qi, L.Q., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Acknowledgements
The research of the first author is supported by NSF of China (No. 11101016). The third author’s research is supported by Scientific Research Common Program of Beijing Municipal Commission of Education (No. KM201210005033) NSF of China (No. 11371001), and China Scholarship Council.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Zhao, XY., Cai, T., Xu, D. (2015). A Newton-CG Augmented Lagrangian Method for Convex Quadratically Constrained Quadratic Semidefinite Programs. In: Gao, D., Ruan, N., Xing, W. (eds) Advances in Global Optimization. Springer Proceedings in Mathematics & Statistics, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-08377-3_33
Download citation
DOI: https://doi.org/10.1007/978-3-319-08377-3_33
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08376-6
Online ISBN: 978-3-319-08377-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)