Roundoff-error-free basis updates of lu factorizations for the efficient validation of optimality certificates

The roundoff-error-free (REF) LU and Cholesky factorizations, combined with the REF substitution algorithms, allow rational systems of linear equations to be solved exactly and efficiently by working entirely in integer arithmetic. These REF computational tools share two key properties: their constituent divisions are exact, and their matrix entries’ bit-lengths (i.e., required number of storage bits) are bounded polynomially. However, as this paper explains, updating the REF factorizations via the common delete-insert-reduce approach turns out to be inefficient in terms of operand bit-length growth and increased computational effort. In fact, we prove that this inefficiency can eliminate the computational savings expected of the factorization update process. Hence, the current work develops REF update algorithms that differ significantly from their traditional counterparts. To achieve this, it introduces the frame matrix methodology for representing and applying the REF factorizations, and then it devises new REF operations and shortcuts for transitioning between “nearby” frame matrices. The presented push-and-swap update approach utilizes these transitions to preserve the special structure of the REF factorizations and to avoid additional growth in their matrix entries’ bit-lengths; this update process requires the expected O(n²) operations. The featured updates are column addition, deletion, and replacement with respect to the REF LU factorization. We also prove that analogous row updates can be performed via the column updates and discuss special considerations for updating the REF Cholesky factorization. An accompanying set of computational tests attests that for fully dense random instances, the push-and-swap column replacement update for the REF LU factorization is on average an order of magnitude faster than the corresponding delete-insert-reduce update for the exact rational arithmetic LU factorization. Altogether, the REF update algorithms represent a promising preliminary step toward enhancing the basic solution validation subroutines employed by state-of-the-art mixed-precision methodologies for exact linear and mixed-integer programming.