Regularization and preconditioning of KKT systems arising in nonnegative least-squares problems

Numerical Linear Algebra with Applications

Volumn 16, Issue 1, 2009, Pages 39-61

  Bellavia, Stefania ^a   Gondzio, Jacek ^b   Morini, Benedetta ^a  

^a UNIVERSITY OF FLORENCE   (Italy)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Bound constrained linear least squares problems; Inexact newton methods; Iterative linear solvers; KKT systems; Preconditioning; Regularization

Indexed keywords

EID: 58449098704     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.610     Document Type: Article

References (23)

1. Björck A. Numerical Methods for Least Squares Problems. SIAM: Philadelphia, PA, 1996.
2. Lotstedt P. Perturbation bounds for the linear least-squares problem subject to linear inequality constraints. BIT 1983; 23:500-519.
3. Bellavia S, Macconi M, Morini B. An interior Newton-like method for nonnegative least-squares problems with degenerate solution. Numerical Linear Algebra with Applications 2006; 13:825-846.
4. Heinkenschloss M, Ulbrich M, Ulbrich S. Superlinear and quadratic convergence of affine-scaling interior-point Newton methods for problems with simple bounds without strict complementarity assumptions. Mathematical Programming 1999; 86:615-635.
5. Saunders M. Cholesky-based methods for sparse least-squares: the benefits of regularization. In Linear and Nonlinear Conjugate Gradient-related Methods, Adams L, Nazareth L (eds). SIAM: Philadelphia, PA, 1996; 92-100.
6. Alunan A, Gondzio J. Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization. Optimization Methods and Software 1999; 11:275-302.
7. Benzi M, Golub GH, Liesen J. Numerical solution of saddle point problems. Acta Numerica 2005; 14:1-137.
8. Dollar HS, Gould NIM, Schilders WHA, Wathen AJ. Implicit-factorization preconditioning and iterative solvers for regularized saddle-point systems. SIAM Journal on Matrix Analysis and Applications 2006; 28:170-189.
9. Coleman TF, Li Y. An interior trust-region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization 1996; 6:418-445.
10. Dembo RS, Eisenstat SC, Steihaug T. Inexact Newton methods. SIAM Journal on Numerical Analysis 1982; 19:400-408.
11. Silvester D, Wathen A. Fast iterative solution of stabilised stokes systems. Part II: using general block preconditioners. SIAM Journal on Numerical Analysis 1994; 31:1352-1367.
12. Saunders M. Solution of sparse rectangular systems using LSQR and CRAIG. BIT 1995; 35:588-604.
13. van der Vorst HA. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonlinear systems. SIAM Journal on Scientific and Statistical Computing 1992; 13:631-644.
14. Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1986; 7:856-869.
15. Freund R, Nachtigal N. Software for simplified Lanczos and QMR algorithms. Applied Numerical Mathematics 1995; 19:319-341.
16. Dollar S. Iterative Linear Algebra for Constrained Optimization. PhD. Thesis, Oxford University Computing Laboratory, Keble College, University of Oxford, 2005.
17. Dollar HS, Gould NIM, Schilders WAH, Wathen AJ. Using constraint preconditioners with regularized saddle-point problems. Computational Optimization and Applications 2007; 36:249-270.
18. Duff I, Grimes R, Lewis J. Sparse matrix test problems. ACM Transactions on Mathematical Software 1989; 15:1-14.
19. Arioli M, Duff IS, de Rijk PPM. On the augmented system approach to sparse least-squares problems. Numerische Mathematik 1989; 55:667-684.
20. Davis TA. The University of Florida sparse matrix collection. NA Digest 1994; 92(42); 1996;
21. 96(28); 1997;
22. 97(23). Available from: http://www.cise.ufl.edu/research/sparse/ matrices.
23. Gondzio J. HOPDM (version 2.12) - a fast LP solver based on a primal-dual interior point method. European Journal of Operational Research 1995; 85:221-225.