Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods

Abstract We devise a hybrid approach for solving linear systems arising from interior point methods applied to linear programming problems. These systems are solved by preconditioned conjugate gradient method that works in two phases. During phase I it uses a kind of incomplete Cholesky preconditioner such that fill-in can be controlled in terms of available memory. As the optimal solution of the problem is approached, the linear systems becomes highly ill-conditioned and the method changes to phase II. In this phase a preconditioner based on the LU factorization is found to work better near a solution of the LP problem. The numerical experiments reveal that the iterative hybrid approach works better than Cholesky factorization on some classes of large-scale problems.

[1]  Andrew J. Wathen,et al.  On implicit-factorization constraint preconditioners , 2006 .

[2]  Luca Bergamaschi,et al.  Preconditioning Indefinite Systems in Interior Point Methods for Optimization , 2004, Comput. Optim. Appl..

[3]  Nicholas I. M. Gould,et al.  On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization , 2001, SIAM J. Sci. Comput..

[4]  Manfred Padberg,et al.  Location, Scheduling, Design and Integer Programming , 2011, J. Oper. Res. Soc..

[5]  Dianne P. O'Leary,et al.  Adaptive use of iterative methods in predictor–corrector interior point methods for linear programming , 2000, Numerical Algorithms.

[6]  J. Gondzio.  HOPDM (version 2.12) — A fast LP solver based on a primal-dual interior point method , 1995 .

[7]  N. Munksgaard,et al.  Solving Sparse Symmetric Sets of Linear Equations by Preconditioned Conjugate Gradients , 1980, TOMS.

[8]  Mauricio G. C. Resende,et al.  Data Structures and Programming Techniques for the Implementation of Karmarkar's Algorithm , 1989, INFORMS J. Comput..

[9]  Frederico F. Campos,et al.  An Efficient Solver for Multi-Right-Hand-Side Linear Systems Based on the CCCG(η) Method with Applications to Implicit Time-Dependent Partial Differential Equations , 1998, SIAM J. Sci. Comput..

[10]  Jacek Gondzio,et al.  Multiple centrality corrections in a primal-dual method for linear programming , 1996, Comput. Optim. Appl..

[11]  Roy E. Marsten,et al.  On Implementing Mehrotra's Predictor-Corrector Interior-Point Method for Linear Programming , 1992, SIAM J. Optim..

[12]  Venansius Baryamureeba.  Solution of large-scale weighted least-squares problems , 2002, Numer. Linear Algebra Appl..

[13]  G. Forsythe,et al.  On best conditioned matrices , 1955 .

[14]  Nicholas I. M. Gould,et al.  Constraint Preconditioning for Indefinite Linear Systems , 2000, SIAM J. Matrix Anal. Appl..

[15]  Stephen J. Wright,et al.  PCx: an interior-point code for linear programming , 1999 .

[16]  L. Portugal,et al.  A truncated primal‐infeasible dual‐feasible network interior point method , 2000 .

[17]  Mauricio G. C. Resende,et al.  An Implementation of the Dual Affine Scaling Algorithm for Minimum-Cost Flow on Bipartite Uncapacitated Networks , 1993, SIAM J. Optim..

[18]  John C. Haws.  Preconditioning KKT Systems , 2002 .

[19]  Mauricio G. C. Resende,et al.  An implementation of Karmarkar's algorithm for linear programming , 1989, Math. Program..

[20]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[21]  Franz Rendl,et al.  QAPLIB – A Quadratic Assignment Problem Library , 1997, J. Glob. Optim..

[22]  A. George,et al.  An Implementation of Gaussian Elimination with Partial Pivoting for Sparse Systems , 1985 .

[23]  Iain S. Duff,et al.  Users' guide for the Harwell-Boeing sparse matrix collection (Release 1) , 1992 .

[24]  Michael A. Saunders,et al.  Preconditioners for Indefinite Systems Arising in Optimization , 1992, SIAM J. Matrix Anal. Appl..

[25]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[26]  C. Durazzi,et al.  Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems , 2003, Numer. Linear Algebra Appl..

[27]  J. Bunch,et al.  Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations , 1971 .

[28]  J. Gondzio,et al.  Regularized Symmetric Indefinite Systems in Interior Point Methods for Linear and Quadratic Optimization , 1999 .

[29]  D. Sorensen,et al.  A new class of preconditioners for large-scale linear systems from interior point methods for linear programming , 2005 .

[30]  Mark T. Jones,et al.  An improved incomplete Cholesky factorization , 1995, TOMS.

[31]  T. Manteuffel.  An incomplete factorization technique for positive definite linear systems , 1980 .

[32]  S. E. Karisch,et al.  QAPLIB-A quadratic assignment problem library , 1991 .

[33]  Ill-conditionedness and Interior-point Methods , 2001 .