Identification, Assessment, and Correction of Ill-Conditioning and Numerical Instability in Linear and Integer Programs

Abstract

The implementation of linear programming (LP) and mixed-integer programming (MIP) algorithms on finite precision computers can create numerical challenges that are not addressed in the mathematical descriptions of these algorithms given in many introductory and more advanced textbooks and courses. Rounding errors associated with finite precision can be magnified because of ill-conditioning or numerical instability, resulting in unexpected, possibly inconsistent results. This tutorial helps the optimization practitioner identify sources of ill-conditioning and numerical instability, assess the cause, and take appropriate remedial action. After discussing some finite precision computing fundamentals, it considers different measures of ill-conditioning, each one of which provides the simplest explanation of ill-conditioning on certain types of LP and MIP models. We then consider remedies for these numerical challenges: (i) optimizer parameter settings that treat the symptoms and (ii) diagnostic tactics that resolve the underlying MIP or LP issue.