## A Hybrid Branch-and-Bound Approach for Exact Rational Mixed-Integer Programming

Please always quote using this URN: urn:nbn:de:0297-zib-17171
• We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the MIPLIB and Mittelmann libraries and for a new collection of numerically difficult instances.

Author: William Cook, Thorsten Koch, Daniel E. Steffy, Kati Wolter ZIB-Report 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C10 Integer programming 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C11 Mixed integer programming 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut 2012/12/12 ZIB-Report (12-49) 1438-0064 Appeared in: Mathematical Programming Computation 5 (2013) 305-344 http://dx.doi.org/10.1007/s12532-013-0055-6

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