## Improving the Accuracy of Linear Programming Solvers with Iterative Refinement

Please always quote using this URN: urn:nbn:de:0297-zib-15451
• 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, right-hand 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 demonstrate that this algorithm is effective in practice for computing extended precision solutions and that this leads to direct improvement of the best known methods for solving LPs exactly over the rational numbers.

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Author: Ambros M. Gleixner, Daniel Steffy, Kati Wolter ZIB-Report Exact linear programming; Iterative refinement; Linear programming 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C05 Linear programming G. Mathematics of Computing / G.4 MATHEMATICAL SOFTWARE D. Software / D.2 SOFTWARE ENGINEERING (K.6.3) / D.2.8 Metrics (D.4.8) I. Computing Methodologies / I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION (REVISED) / I.1.2 Algorithms (F.2.1-2) 2012/05/08 ZIB-Report (12-19) 1438-0064 Appeared in: ISSAC '12 Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation pp. 187-194 http://dx.doi.org/10.1145/2442829.2442858

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