Identification, Assessment, and Correction of Ill-Conditioning and Numerical Instability in Linear and Integer Programs

Klotz, Ed

doi:10.1287/educ.2014.0130

Cited by 9 publications

(6 citation statements)

References 14 publications

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“…Several remarks on the reasonableness of Assumption 3.5 are in order. As Klotz (2014) points out for the CPLEX LP code, state-of-the-art LP solvers typically use an absolute definition for their tolerance requirements. First and foremost, because limited floatingpoint precision is by construction relative, an LP solver based only on floating-point computation will in general not be able to return solutions within absolute tolerances-certainly not the fast LP solvers we have in mind for practical applications.…”

Section: Convergencementioning

confidence: 99%

Iterative Refinement for Linear Programming

Gleixner

Steffy

Wolter³

2016

INFORMS Journal on Computing

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We describe an iterative refinement procedure for computing extended precision or exact solutions to linear programming problems (LPs). Arbitrarily precise solutions can be computed by solving a sequence of closely related LPs with limited precision arithmetic. The LPs solved share the same constraint matrix as the original problem instance and are transformed only by modification of the objective function, righthand side, and variable bounds. Exact computation is used to compute and store the exact representation of the transformed problems, while numeric computation is used for solving LPs. At all steps of the algorithm the LP bases encountered in the transformed problems correspond directly to LP bases in the original problem description. We show that this algorithm is effective in practice for computing extended precision solutions and that it leads to a direct improvement of the best known methods for solving LPs exactly over the rational numbers. Our implementation is publically available as an extension of the academic LP solver SoPlex.

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Section: Convergencementioning

confidence: 99%

Iterative Refinement for Linear Programming

Gleixner

Steffy

Wolter³

2016

INFORMS Journal on Computing

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“…Exactly solving sparse systems of linear equations (SLEs) is a key subroutine of algorithms used to solve problems arising in various ields including number theory [Dixon 1982;Wiedemann 1986], mathematical proofs [Hales 2005], computational geometry [Burton and Ozlen 2012;Gärtner 1999], and exact linear/integer programming [Gleixner 2015;Stefy 2011]. Moreover, solving a sparse SLE may require extended precision due to numerical instability [Higham 2002;Klotz 2014] or a poorly scaled or highly ill-conditioned input matrix [Golub and Van Loan 2012;Horn and Johnson 2012]. Lourenco et al [2019] derived the sparse left-looking integer-preserving (SLIP) LU factorization, which exactly solves the sparse SLE, Ax = b exclusively using integer-arithmetic; thereby ensuring that the inal solution to the system is roundof-error-free.…”

Section: Overviewmentioning

confidence: 99%

Algorithm 1021: SPEX Left LU, Exactly Solving Sparse Linear Systems via a Sparse Left-looking Integer-preserving LU Factorization

Lourenco

Chen

Moreno-Centeno

et al. 2022

ACM Trans. Math. Softw.

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SPEX Left LU is a software package for exactly solving unsymmetric sparse linear systems. As a component of the sparse exact (SPEX) software package, SPEX Left LU can be applied to any input matrix, A , whose entries are integral, rational, or decimal, and provides a solution to the system Ax = b which is either exact or accurate to user-specified precision. SPEX Left LU preorders the matrix A with a user-specified fill-reducing ordering and computes a left-looking LU factorization with the special property that each operation used to compute the L and U matrices is integral. Notable additional applications of this package include benchmarking the stability and accuracy of state-of-the-art linear solvers, and determining whether singular-to-double-precision matrices are indeed singular. Computationally, this paper evaluates the impact of several novel pivoting schemes in exact arithmetic, benchmarks the exact iterative solvers within Linbox, and benchmarks the accuracy of MATLAB sparse backslash. Most importantly, it is shown that SPEX Left LU outperforms the exact iterative solvers in run time on easy instances and in stability as the iterative solver fails on a sizeable subset of the tested (both easy and hard) instances. The SPEX Left LU package is written in ANSI C, comes with a MATLAB interface, and is distributed via GitHub, as a component of the SPEX software package, and as a component of SuiteSparse.

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“…As frequently described in the literature, too wide a range of numbers inside a single MIP model is likely to deteriorate the numeric stability and significantly reduce the solver performance (see for example [3]).…”

Section: Numerical Stabilitymentioning

confidence: 99%

Tackling Industrial-Scale Supply Chain Problems by Mixed-Integer Programming

sci¹

2019

JCM

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The modeling flexibility and the optimality guarantees provided by mixed-integer programming greatly aid the design of robust and future-proof decision support systems. The complexity of industrial-scale supply chain optimization, however, often poses limits to the application of general mixed-integer programming solvers. In this paper we describe algorithmic innovations that help to ensure that MIP solver performance matches the complexity of the large supply chain problems and tight time limits encountered in practice. Our computational evaluation is based on a diverse set, modeling real-world scenarios supplied by our industry partner SAP.

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Identification, Assessment, and Correction of Ill-Conditioning and Numerical Instability in Linear and Integer Programs

Cited by 9 publications

References 14 publications

Iterative Refinement for Linear Programming

Iterative Refinement for Linear Programming

Algorithm 1021: SPEX Left LU, Exactly Solving Sparse Linear Systems via a Sparse Left-looking Integer-preserving LU Factorization

Tackling Industrial-Scale Supply Chain Problems by Mixed-Integer Programming

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