Logarithmic barriers for sparse matrix cones

Optimization Methods and Software

Volumn 28, Issue 3, 2013, Pages 396-423

  Andersen, Martin S ^a   Dahl, Joachim ^b   Vandenberghe, Lieven ^a  

^a Engineering IV   (United States)

^b MOSEK ApS ^*  (Denmark)

Author keywords

multifrontal method; semidefinite programming; sparse matrices

Indexed keywords

LARGE-SCALE ALGORITHMS; LOGARITHMIC BARRIER FUNCTION; MULTIFRONTAL METHODS; POSITIVE SEMIDEFINITE MATRICES; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE PROGRAMS; SPARSE CHOLESKY FACTORIZATIONS; SPARSE MATRICES;

ALGORITHMS; CONES; MATHEMATICAL PROGRAMMING;

MATRIX ALGEBRA;

EID: 84878327557     PISSN: 10556788     EISSN: 10294937     Source Type: Journal    
DOI: 10.1080/10556788.2012.684353     Document Type: Article

References (40)

1. P.R. Amestoy, I.S. Duff, and J. L'Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Eng. 184 (2000), pp. 501-520.
2. P.R. Amestoy, I.S. Duff, Y. Robert, F.H. Rouet, and B. Uçar, On computing inverse entries of a sparse matrix in an out-of-core environment, Tech. Rep. TR/PA/10/59, CERFACS, Toulouse, 2010.
3. M.S. Andersen, J. Dahl, and L. Vandenberghe, Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones, Math. Program. Comput. 2 (2010), pp. 167-201.
4. C. Ashcraft and R. Grimes, The influence of relaxed supernode partitions on the multifrontal method, ACM Trans. Math. Softw. 15 (1989), pp. 291-309.
5. O. Banerjee, L. El Ghaoui, and A. d'Aspremont, Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data, J. Mach. Learn. Res. 9 (2008), pp. 485-516.
6. J.R.S. Blair and B. Peyton, An introduction to chordal graphs and clique trees, in Graph Theory and Sparse Matrix Computation, A. George, J.R. Gilbert, and J.W.H. Liu, eds., Springer-Verlag, NewYork, 1993, pp. 1-29.
7. S. Burer, Semidefinite programming in the space of partial positive semidefinite matrices, SIAM J. Optim. 14 (2003), pp. 139-172.
8. Y.E. Campbell and T.A. Davis, Computing the sparse inverse subset: An inverse multifrontal approach, Tech. Rep. TR-95-021, Computer and Information Sciences Department, University of Florida, Gainesville, FL, 1995.
9. Y. Chen, T.A. Davis, W.W. Hager, and S. Rajamanickam, Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate, ACM Trans. Math. Softw. 35 (2008), pp. 1-14.
10. J. Dahl and L.Vandenberghe, CHOMPACK: Chordal Matrix Package, 2009; available at abel.ee.ucla.edu/chompack.
11. J. Dahl and L. Vandenberghe, CVXOPT: A Python Package for Convex Optimization, 2010; available at abel.ee.ucla.edu/cvxopt.
12. J. Dahl, L. Vandenberghe, and V. Roychowdhury, Covariance selection for non-chordal graphs via chordal embedding, Optim. Methods Softw. 23 (2008), pp. 501-520.
13. A. d'Aspremont, O. Banerjee, and L. El Ghaoui, First-order methods for sparse covariance selection, SIAM J. Matrix Anal. Appl. 30 (2008), pp. 56-66.
14. T.A. Davis. Direct Methods for Sparse Linear Systems. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2006.
15. T.A. Davis, The University of Florida Sparse Matrix Collection, Tech. Rep., Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL, 2009; available at http://www.cise. ufl.edu/research/sparse/matrices.
16. J.J. Dongarra, J.D. Croz, S. Hammarling, and R.J. Hanson, An extended set of Fortran basic linear algebra subprograms, ACM Trans. Math. Softw. 14 (1988), pp. 1-17.
17. J.J. Dongarra, J.D. Croz, S. Hammarling, and I. Duff, A set of level 3 basic linear algebra subprograms, ACMTrans. Math. Softw. 16 (1990), pp. 1-17.
18. I.S. Duff and J.K. Reid, The multifrontal solution of indefinite sparse symmetric linear equations, ACMTrans. Math. Softw. 9 (1983), pp. 302-325.
19. R. Fletcher, An optimal positive definite update for sparse Hessian matrices, SIAM J. Optim. 5 (1995), pp. 192-218.
20. J. Friedman, T. Hastie, and R. Tibshirani, Sparse inverse covariance estimation with the graphical lasso, Biostatistics 9 (2008), pp. 432-441.
21. M. Fukuda, M. Kojima, K. Murota, and K. Nakata, Exploiting sparsity in semidefinite programming via matrix completion I: General framework, SIAM J. Optim. 11 (2000), pp. 647-674.
22. G.H. Golub and R.J. Plemmons, Large-scale geodetic least-squares adjustment by dissection and orthogonal decomposition, Linear Algebra Appl. 34 (1980), pp. 3-27.
23. G.H. Golub and C.F.Van Loan. Matrix Computations. 3rd edition, Johns Hopkins University Press, Baltimore, MD, 1996.
24. A. Griewank and A.Walther. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. 2nd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008.
25. R. Grone, C.R. Johnson, E.M. Sá, and H. Wolkowicz, Positive definite completions of partial Hermitian matrices, Linear Algebra Appl. 58 (1984), pp. 109-124.
26. A. Guermouche and J.Y. L'Excellent, Constructing memory-minimizing schedules for multifrontal methods, ACM Trans. Math. Softw. 32 (2006), pp. 17-32.
27. J.Z. Huang, N. Liu, M. Pourahmadi, and L. Liu, Covariance matrix selection and estimation via penalised normal likelihood, Biometrika 93 (2006), pp. 85-98.
28. M. Laurent, Matrix completion problems, in Encyclopedia of Optimization III, C.A. Floudas and P.M. Pardalos, eds., Kluwer, Dordrecht, 2001, pp. 221-229.
29. J.G. Lewis, B.W. Peyton, and A. Pothen, A fast algorithm for reordering sparse matrices for parallel factorization, SIAM J. Sci. Statist. Comput. 10 (1989), pp. 1146-1173.
30. L. Li and K.C. Toh, An inexact interior point method for L1-regularized sparse covariance selection, Math. Program. Comput. 2 (2010), pp. 291-315.
31. J.W.H. Liu, On the storage requirement in the out-of-core multifrontal method for sparse factorization, ACM Trans. Math. Softw. 12 (1986), pp. 249-264.
32. J.W.H. Liu, Equivalent sparse matrix reordering by elimination tree rotations, SIAM J. Sci. Statist. Comput. 9 (1988), pp. 424-444.
33. J.W.H. Liu, The role of elimination trees in sparse factorization, SIAM J. Matrix Anal. Appl. 11 (1990), pp. 134-172.
34. J.W.H. Liu, The multifrontal method for sparse matrix solution: Theory and practice, SIAM Rev. 34 (1992), pp. 82-109.
35. J.W.H. Liu, E.G. Ng, and B.W. Peyton, On finding supernodes for sparse matrix computations, SIAM J. Matrix Anal. Appl. 14 (1993), pp. 242-252.
36. K. Nakata, K. Fujitsawa, M. Fukuda, M. Kojima, and K. Murota, Exploiting sparsity in semidefinite programming via matrix completion II: Implementation and numerical details, Math. Program. Ser. B 95 (2003), pp. 303-327.
37. A. Pothen and C. Sun, Compact clique tree data structures in sparse matrix factorizations, in Large-Scale Numerical Optimization, T.F. Coleman andY. Li, eds., Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990, pp. 180-204.
38. K. Scheinberg, S. Ma, and D. Goldfarb, Sparse inverse covariance selection via alternating linearization methods, in Advances in Neural Information Processing Systems 23, J. Lafferty, C.K.I.Williams, J. Shawe-Taylor, R. Zemel, and A. Culotta, eds., Curran Associates, Inc., Red Hook, NY, 2010, pp. 2101-2109.
39. G. Srijuntongsiri and S.A. Vavasis, A Fully Sparse Implementation of a Primal-Dual Interior-Point Potential Reduction Method for Semidefinite Programming, 2004; available at http://arxiv.org/abs/cs/0412009.
40. N. Yamashita, Sparse quasi-Newton updates with positive definite matrix completion, Math. Program. Ser. A 115 (2008), pp. 1-30.