G.4 MATHEMATICAL SOFTWARE
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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, 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.
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 collections.