Efficiently computable precision and tolerance bounds

Determine efficiently computable non-trivial bounds for the problem-dependent tolerance threshold τ⋆ and floating-point unit-roundoff threshold ϵ⋆ required for the tolerance-aware floating-point simplex algorithm to reproduce the exact simplex decisions on a given linear-programming instance.

Background

The paper proves that, for every standard rational linear-programming instance with a feasible starting basis, sufficiently small problem-dependent tolerance and unit-roundoff parameters ensure that the floating-point simplex algorithm using the Bartels–Golub basis update makes the same optimality, unboundedness, entering-variable, and leaving-variable decisions as the exact rational simplex algorithm.

The required thresholds τ⋆ and ϵ⋆ depend on quantities such as the least negative reduced cost, the smallest positive entering-variable coefficient, and the smallest gap between distinct ratio-test values across all admissible bases. The paper establishes their existence but states that no efficiently computable non-trivial bounds are known. Such bounds would support principled parameter selection instead of the heuristic strategy of repeatedly decreasing the tolerance and precision until the computation succeeds.