Foundational Factorization Algorithms for the Efficient Roundoff-Error-Free Solution of Optimization Problems

LU and Cholesky factorizations play a central role in solving linear and mixedinteger programs. In many documented cases, the roundoff errors accrued during the construction and implementation of these factorizations cause the misclassification of suboptimal solutions as optimal and infeasible problems as feasible and viceversa. Such erroneous outputs bring the reliability of optimization solvers into question and, therefore, it is imperative to eliminate these roundoff errors altogether and to do so efficiently to ensure practicality. Firstly, this work introduces two roundoff-error-free factorizations (REF) constructed exclusively in integer arithmetic: the REF LU and Cholesky factorizations. Additionally, it develops supplementary integer-preserving substitution algorithms, thereby providing a complete tool set for solving systems of linear equations (SLEs) exactly and efficiently. An inherent property of the REF factorization algorithms is that their entries’ bit-length—i.e., the number of bits required for expression—is bounded polynomially. Unlike the exact rational arithmetic methods used in practice, however, the algorithms herein presented do not require any greatest common divisor operations to guarantee this pivotal property. Secondly, this work derives various useful theoretical results and details computational tests to demonstrate that the REF factorization framework is considerably superior to the rational arithmetic LU factorization approach in computational performance and storage requirements. This is significant because the latter approach is the solution validation tool of choice of state-of-the-art exact linear programming solvers due to its ability to handle both numerically difficult and intricate problems. An additional theoretical contribution and further computational tests also demon-