Abstract
The interior-point method solves large linear programming problems in a few iterations. Each iteration requires computing the solution to one or more linear systems. This constitutes the most expensive step of the method and reducing the time to solve these linear systems is a way of improving the method’s performance. Iterative methods such as the preconditioned conjugate gradient method can be used to solve them. Incomplete Cholesky factorization can be used as a preconditioner to the problem. However, breakdowns may occur during incomplete factorizations and corrections on the diagonal may be required. This correction is accomplished by adding a positive number to diagonal elements of the linear system matrix and the factorization of the new matrix is restarted. In this work, we propose modifications to controlled Cholesky factorization to avoid or decrease the number of refactorizations of diagonally modified matrices. Computational results show that the proposed techniques can reduce the time needed for solving linear programming problems by the interior-point method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellavia S, de Simone V, di Serafino D, Morini B (2011) Efficient preconditioner updates for shifted linear systems. SIAM J Sci Comput 4:1785–1809
Bellavia S, de Simone V, di Serafino D, Morini B (2012) A preconditioning framework for sequences of diagonally modified linear systems arising in optimization. SIAM J Numer Anal 6:3280–3302
Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137
Bergamaschi L, Gondzio J, Venturin Mand Zilli G (2007) Inexact constraint preconditioners for linear system arising interior point methods. Comput Optim Appl 36:137–147
Bocanegra S, Campos FF, Oliveira ARL (2007) Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods 36:149–164
Burkard RE, Karisch S, Rendl F (1991) QAPLIB-A quadratic assignment problem library. Eur J Oper Res 55:115–119
Campos FF, Birkett NRC (1998) An efficient solver for multi-right hand side linear systems based on the CCCG(\(\eta \)) method with applications to implicit time-dependent partial differential equations, SIAM. J Sci Comput 19:126–138
Czyzyk J, Mehrotra S, Wagner M, Wright SJ (1999) PCx: an interior-point code for linear programming. Optim Methods Softw 11:397–430
Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math Program 91:201–213
Ghidini CTLS, Oliveira ARL, Silva J, Velazco MI (2012) Combining a hybrid preconditioner and a optimal adjustment algorithm to accelerate the convergence of interior point methods. Linear Algebra Appl 436:1267–1284
Ghidini CTLS, Oliveira ARL, Sorensen DC (2014) Computing a hybrid preconditioner approach to solve the linear systems arising from interior point methods for linear programming using the conjugate gradient method. Ann Manag Sci 3:44–66
Gondzio J (2012) Interior point methods 25 years later. Eur J Oper Res 218:587–601
Jones MT, Plassmann PE (1995) An improved incomplete Cholesky factorization. ACM Tran Math Softw 21:5–17
Kershaw DS (1978) The incomplete Cholesky - conjugate gradient method for the iterative solution of systems of linear equations. J Comput Phys 26:43–65
Manteuffel TA (1980) An incomplete factorization technique for positive definite linear systems. Math Comput 34:473–497
Mehrotra S (1992) On the implementation of a primal-dual interior point method. SIAM J Optim 2:575–601
Oliveira ARL, Sorensen DC (2005) A new class of preconditioners for large-scale linear systems from interior point methods for linear programming. Linear Algebra Appl 394:1–24
Resende MGC, Veiga G (1993) An efficient implementation of a network interior point method. DIMACS Ser Discr Math Theoret Comput Sci 12:299–348
Velazco MI, Oliveira ARL, Campos FF (2010) A note on hybrid preconditioners for large-scale normal equations arising from interior-point methods. Optim Methods Softw 25:321–332
Wright SJ (1997) Primal-Dual Interior-Point Methods, 289. Society for Industrial and Applied Mathematics, Philadelphia
Acknowledgements
The authors would like to thank the Brazilian institutions CAPES, CNPq, and FAPESP for financial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Joerg Fliege.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Silva, L.M., Oliveira, A.R.L. Modified controlled Cholesky factorization for preconditioning linear systems from the interior-point method. Comp. Appl. Math. 40, 154 (2021). https://doi.org/10.1007/s40314-021-01544-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-021-01544-0