A newton-cg augmented lagrangian method for semidefinite programming

SIAM Journal on Optimization

Volumn 20, Issue 4, 2010, Pages 1737-1765

  Zhao, Xin Yuan ^a   Sun, Defeng ^b   Toh, Kim Chuan ^b  

^a BEIJING UNIVERSITY OF TECHNOLOGY   (China)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Augmented lagrangian; Iterative solver; Newton method; Semidefinite programming; Semismoothness

Indexed keywords

AUGMENTED LAGRANGIANS; ITERATIVE SOLVERS; NEWTON METHODS; SEMI-DEFINITE PROGRAMMING; SEMISMOOTHNESS;

APPROXIMATION THEORY; LAGRANGE MULTIPLIERS; NEWTON-RAPHSON METHOD;

MATHEMATICAL PROGRAMMING;

EID: 77951274410     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/080718206     Document Type: Article

References (45)

1. F. Alizadeh, J. P. A. Haeberly, and O. L. Overton, Complementarity and nondegeneracy in semidefinite programming, Math. Programming, 77 (1997), pp. 111-128.
2. V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp.29-43.
3. Z.-J. Bai, D. Chu, and D. Sun, A dual optimization approach to inverse quadratic eigenvalueproblems with partial eigenstructure, SIAM J. Sci. Comput., 29 (2007), pp. 2531-2561.
4. J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
5. S. Burer and D. Vandenbussche, Solving lift-and-project relaxations of binary integer programs, SIAM J. Optim., 16 (2006), pp. 726-750.
6. S. Burer, R. Monteiro, and Y. Zhang, A computational study of a gradient-based log-barrier algorithm for a class of large scale SDPs, Math. Program., 95 (2003), pp. 359-379.
7. Z. X. Chan and D. Sun, Constraint nondegeneracy, strong regularity, and nonsingularity in semidefinite programming, SIAM J. Optim., 19 (2008), pp. 370-396.
8. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
9. A. Eisenblätter, M. Grötschel, and A. M. C. A. Koster, Frequency planning and ramification of coloring, Discuss. Math. Graph Theory, 22 (2002), pp. 51-88.
10. F. Facchinei, Minimization of SC1 functions and the Maratos effect, Oper. Res. Lett., 17 (1995), pp. 131-137.
11. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, London, 1996.
12. C. Helmberg, Numerical Evaluation of SB method, Math. Program., 95 (2003), pp. 381-406.
13. F. Jarre and F. Rendl, An augmented primal-dual method for linear conic programs, SIAM J. Optim., 19 (2008), pp. 808-823.
14. C. Jansson, Termination and Verification for Ill-Posed Semidefinite Programs, Technical report, Informatik III, TU Hamburg-Harburg, Hamburg, Germany, 2005.
15. D. Johnson, G. Pataki, and Farid Alizadeh, The Seventh DIMACS Implementation Challenge: Semidefinite and Related Optimization Problems, Rutgers University, New Brunswick, NJ, 2000; available online at
16. M. Kočvara and M. Stingl, PENNON: A code for convex nonlinear and semidefinite programming, Optim. Methods Softw., 18 (2003), pp. 317-333.
17. M. Kočvara and M. Stingl, On the solution of large-scale SDP problems by the modified barrier method using iterative solvers, Math. Program., 109 (2007), pp. 413-444.
18. B. Kummer, Newton's method for nondifferentiable functions, in Advances in Mathematical Optimization, J. Guddat et al., eds., Akademie-Verlag, Berlin, 1988, pp. 114-125.
19. J. Malick, J. Povh, F. Rendl, and A. Wiegele, Regularization methods for semidefinite programming, SIAM J. Optim., 20 (2009), pp. 336-356.
20. H. Mittelmann, An independent benchmarking of SDP and SOCP solvers, Math. Program., 95 (2003), pp. 407-430.
21. J. S. Pang and L. Qi, A globally convergent Newton method of convex SC1 minimization problems, J. Optim. Theory Appl., 85 (1995), pp. 633-648.
22. J. S. Pang, D. F. Sun, and J. Sun, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Math. Oper. Res., 28 (2003), pp. 272-284.
23. J. Povh and F. Rendl, Copositive and semidefinite relaxations of the quadratic assignment problem, Discrete Optim., 6 (2009), pp. 231-241.
24. J. Povh, F. Rendl, and A. Wiegele, A boundary point method to solve semidefinite programs, Computing, 78 (2006), pp. 277-286.
25. H. Qi and D. Sun, A quadratically convergent Newton method for computing the nearest correlation matrix, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 360-385.
26. L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-367.
27. S. M. Robinson, Local structure of feasible sets in nonlinear programming. II. Nondegeneracy, Math. Programming Stud., No.22 (1984), pp. 217-230.
28. S. M. Robinson, Local structure of feasible sets in nonlinear programming. III. Stability and sensitivity, Math. Programming Stud., No.30 (1987), pp. 45-66.
29. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
30. R. T. Rockafellar, A dual approach to solving nonlinear programming problems by unconstrained optimization, Math. Programming, 5 (1973), pp. 354-373.
31. R. T. Rockafellar, Conjugate Duality and Optimization, CBMS-NSF Regional Conf. Ser. In Appl. Math. 16, SIAM, Philadelphia, 1974.
32. R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
33. R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), pp. 97-116.
34. T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J. Numer. Anal., 20 (1983), pp. 626-637.
35. N. Sloane, Challenge Problems: Independent Sets in Graphs, ~njas/doc/graphs.html.
36. D. F. Sun and J. Sun, Semismooth matrix valued functions, Math. Oper. Res., 27 (2002), pp. 150-169.
37. D. F. Sun, J. Sun, and L. W. Zhang, The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming, Math. Program., 114 (2008), pp. 349-391.
38. L. N. Trefethen and D. Bau, III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
39. K.-C. Toh, Solving large scale semidefinite programs via an iterative solver on the augmented systems, SIAM J. Optim., 14 (2003), pp. 670-698.
40. K. C. Toh, An inexact primal-dual path-following algorithm for convex quadratic SDP, Math. Program., 112 (2007), pp. 221-254.
41. K.-C. Toh and M. Kojima, Solving some large scale semidefinite programs via the conjugate residual method, SIAM J. Optim., 12 (2002), pp. 669-691.
42. M. Trick, V. Chvatal, W. Cook, D. Johnson, C. McGeoch, and R. Tarjan, The Second DIMACS Implementation Challenge: NP Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability, Rutgers University, New Brunswick, NJ, 1992; available online at .
43. A. Wiegele, Biq Mac Library-A Collection of Max-Cut and Quadratic 0-1 Programming Instances of Medium Size, preprint, Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria, 2007.
44. E. H. Zarantonello, Projections on convex sets in Hilbert space and spectral theory. I and II, in Contributions to Nonlinear Functional Analysis, E. H. Zarantonello, ed., Academic Press, New York, 1971, pp. 237-341 and 343-424.
45. X. Y. Zhao, D. Sun, and K. C. Toh, A Newton-CG Augmented Lagrangian Method for Semidefinite Programming, preprint, National University of Singapore, 2008; available online at http://www.math.nus.edu.sg/~mattohkc/papers/SDPNAL. pdf.