Roundoff-Error-Free Basis Updates of LU Factorizations for the Efficient Validation of Optimality Certificates

The roundoff-error-free (REF) LU and Cholesky factorizations, combined with the REF substitution algorithms, allow rational systems of linear equations to be solved exactly and efficiently by worki...

[1]  Erwin H. Bareiss,et al.  Computational Solutions of Matrix Problems Over an Integral Domain , 1972 .

[2]  Kurt Mehlhorn,et al.  Certifying and repairing solutions to large LPs how good are LP-solvers? , 2003, SODA '03.

[3]  Arnold Neumaier,et al.  Safe bounds in linear and mixed-integer linear programming , 2004, Math. Program..

[4]  William J. Cook,et al.  Exact solutions to linear programming problems , 2007, Oper. Res. Lett..

[5]  Kory W. Hedman,et al.  Topology Control for Load Shed Recovery , 2014, IEEE Transactions on Power Systems.

[6]  Douglas H. Wiedemann.  Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[7]  Nikolaos V. Sahinidis,et al.  A review of the LU update in the simplex algorithm , 2012, Int. J. Math. Oper. Res..

[8]  B. David Saunders,et al.  Fraction Free Gaussian Elimination for Sparse Matrices , 1995, J. Symb. Comput..

[9]  Erick Moreno-Centeno,et al.  Roundoff-Error-Free Algorithms for Solving Linear Systems via Cholesky and LU Factorizations , 2015, INFORMS J. Comput..

[10]  Ed Klotz,et al.  Identification, Assessment, and Correction of Ill-Conditioning and Numerical Instability in Linear and Integer Programs , 2014 .

[11]  P. Gill,et al.  Maintaining LU factors of a general sparse matrix , 1987 .

[12]  E. Bareiss.  Sylvester’s identity and multistep integer-preserving Gaussian elimination , 1968 .

[13]  Erich Kaltofen,et al.  Distributed Matrix-Free Solution of Large Sparse Linear Systems over Finite Fields , 1999, Algorithmica.

[14]  Daniel E. Steffy,et al.  Iterative Refinement for Linear Programming , 2016, INFORMS J. Comput..

[15]  Uwe H. Suhl,et al.  A fast LU update for linear programming , 1993, Ann. Oper. Res..

[16]  Zhendong Wan,et al.  An algorithm to solve integer linear systems exactly using numerical methods , 2006, J. Symb. Comput..

[17]  Gene H. Golub,et al.  The simplex method of linear programming using LU decomposition , 1969, Commun. ACM.

[18]  Numerische Mathematik.  Exact Solution of Linear Equations Using P-Adie Expansions* , 2005 .

[19]  John A. Tomlin,et al.  Updated triangular factors of the basis to maintain sparsity in the product form simplex method , 1972, Math. Program..

[20]  William J. Cook,et al.  Solving Very Sparse Rational Systems of Equations , 2011, TOMS.

[21]  J. Edmonds.  Systems of distinct representatives and linear algebra , 1967 .

[22]  William J. Cook,et al.  An Exact Rational Mixed-Integer Programming Solver , 2011, IPCO.

[23]  William J. Cook,et al.  Konrad-zuse-zentrum F ¨ Ur Informationstechnik Berlin a Hybrid Branch-and-bound Approach for Exact Rational Mixed-integer Programming , 2022 .

[24]  Gilles Villard,et al.  Solving sparse rational linear systems , 2006, ISSAC '06.

[25]  Robert A. van de Geijn,et al.  Updating an LU Factorization with Pivoting , 2008, TOMS.

[26]  John K. Reid,et al.  A sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases , 1982, Math. Program..

[27]  M. Saunders.  Large-scale linear programming using the Cholesky factorization , 1972 .

[28]  Thorsten Koch.  The final NETLIB-LP results , 2004, Oper. Res. Lett..