Improving the accuracy of linear programming solvers with iterative refinement

Gleixner, Ambros; Steffy, Daniel E.; Wolter, Kati

doi:10.1145/2442829.2442858

Cited by 34 publications

(38 citation statements)

References 21 publications

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“…19 is devoted to multiscale metabolic networks, showing significant improvement relative to CPLEX5. Our work is complementary and confirms the value of enhancing the simplex solver in refs 17, 18, 19, 20 to employ quadruple-precision computation, as we have done here.…”

supporting

confidence: 82%

“…Here we name the approach DQQ and report experiments with an analogous but cheaper DRR procedure based on conventional iterative refinement of all linear equations arising in the simplex method (see Methods section and Supplementary Information). We also became aware of the work of Gleixner et al 17181920. and their thorough and successful implementation of iterative refinement in SoPlex80 bit.…”

Section: Overviewmentioning

confidence: 99%

“…However, we learned that DRR may lose ground during periodic refactorizations of the simplex basis matrix B , if the current B is nearly singular and “basis repair” becomes necessary. Our DQQ and DRR experience points to the need for an optional Quad version of the basic SoPlex solver to ensure maximum reliability of the refinement approach in refs 17, 18, 19, 20. Meanwhile, DQQ will be effective on a wide range of problems as long as step D finishes naturally or is limited to a reasonable number of iterations before steps Q1 and Q2 take over.…”

Section: Overviewmentioning

confidence: 99%

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Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression

Yang

Fleming

et al. 2017

Sci Rep

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Constraint-Based Reconstruction and Analysis (COBRA) is currently the only methodology that permits integrated modeling of Metabolism and macromolecular Expression (ME) at genome-scale. Linear optimization computes steady-state flux solutions to ME models, but flux values are spread over many orders of magnitude. Data values also have greatly varying magnitudes. Standard double-precision solvers may return inaccurate solutions or report that no solution exists. Exact simplex solvers based on rational arithmetic require a near-optimal warm start to be practical on large problems (current ME models have 70,000 constraints and variables and will grow larger). We have developed a quadruple-precision version of our linear and nonlinear optimizer MINOS, and a solution procedure (DQQ) involving Double and Quad MINOS that achieves reliability and efficiency for ME models and other challenging problems tested here. DQQ will enable extensive use of large linear and nonlinear models in systems biology and other applications involving multiscale data.

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supporting

confidence: 82%

Section: Overviewmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

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Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression

Yang

Fleming

et al. 2017

Sci Rep

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“…The kernel routines run over small finite fields and are usually lifted over Z, Q or Z[X]. They are used in algebraic cryptanalysis [15,3], computational number theory [27], or integer linear programming [18] and they benefit from the experience in numerical linear algebra. In particular, a key point there is to embed the finite field elements in integers stored as floating point numbers, and then rely on the efficiency of the floating point matrix multiplication dgemm of the BLAS.…”

Section: Introductionmentioning

confidence: 99%

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

et al. 2016

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We present block algorithms and their implementation for the parallelization of sub-cubic Gaussian elimination on shared memory architectures. Contrarily to the classical cubic algorithms in parallel numerical linear algebra, we focus here on recursive algorithms and coarse grain parallelization. Indeed, sub-cubic matrix arithmetic can only be achieved through recursive algorithms making coarse grain block algorithms perform more efficiently than fine grain ones. This work is motivated by the design and implementation of dense linear algebra over a finite field, where fast matrix multiplication is used extensively and where costly modular reductions also advocate for coarse grain block decomposition. We incrementally build efficient kernels, for matrix multiplication first, then triangular system solving, on top of which a recursive PLUQ decomposition algorithm is built. We study the parallelization of these kernels using several algorithmic variants: either iterative or recursive and using different splitting strategies. Experiments show that recursive adaptive methods for matrix multiplication, hybrid recursive-iterative methods for triangular system solve and tile recursive versions of the PLUQ decomposition, together with various data mapping policies, provide the best performance on a 32 cores NUMA architecture. Overall, we show that the overhead of modular reductions is more than compensated by the fast linear algebra algorithms and that exact dense linear algebra matches the performance of full rank reference numerical software even in the presence of rank deficiencies.

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“…The most prominent of these alternatives are divided roughly into three categories: p-adic methods, e.g., Dixon (1982), Mulders and Storjohann (2000), Eberly et al (2006); black box linear algebra methods, e.g., Wiedemann (1986), Kaltofen and Saunders (1991), Kaltofen and Lobo (1999); and iterative numerical methods, e.g., Wan (2006), Gleixner et al (2012). Briefly stated, the first two classifications of these major alternative approaches prioritize space complexity, but the third seeks to lower the number of operations performed.…”

Section: Discussionmentioning

confidence: 99%

Roundoff-Error-Free Algorithms for Solving Linear Systems via Cholesky and LU Factorizations

Escobedo

Moreno-Centeno

2015

INFORMS Journal on Computing

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emc@tamu.edu} L U and Cholesky factorizations are computational tools for efficiently solving linear systems that play a central role in solving linear programs and several other classes of mathematical programs. In many documented cases, however, the roundoff errors accrued during the construction and implementation of these factorizations lead to the misclassification of feasible problems as infeasible and vice versa. Hence, reducing these roundoff errors or eliminating them altogether is imperative to guarantee the correctness of the solutions provided by optimization solvers. To achieve this goal without having to use rational arithmetic, we introduce two roundofferror-free factorizations that require storing the same number of individual elements and performing a similar number of operations as the traditional LU and Cholesky factorizations. Additionally, we present supplementary roundoff-error-free forward and backward substitution algorithms, thereby providing a complete tool set for solving systems of linear equations exactly and efficiently. An important property shared by the featured factorizations and substitution algorithms is that their individual coefficients' maximum word length-i.e., the maximum number of digits required for expression-is bounded polynomially. Unlike the rational arithmetic methods used in practice to solve linear systems exactly, however, the algorithms herein presented do not require any gcd calculations to bound the entries' word length. We also derive various other related theoretical results, including the total computational complexity of all the roundoff-error-free processes herein presented.

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Improving the accuracy of linear programming solvers with iterative refinement

Cited by 34 publications

References 21 publications

Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression

Reliable and efficient solution of genome-scale models of Metabolism and macromolecular Expression

Recursion based parallelization of exact dense linear algebra routines for Gaussian elimination

Roundoff-Error-Free Algorithms for Solving Linear Systems via Cholesky and LU Factorizations

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