Tolerance analysis in linear systems and linear programming

Abstract

In many practical linear programming problems, it is often important to know how different optimality criteria (optimal solution, optimal basis, optimal partition, etc.) change under input data perturbations. Our aim is to compute tolerances (intervals) for the objective function and the right-hand side coefficients such that these coefficients can independently and simultaneously vary inside their tolerances while preserving the corresponding optimality criterion. We put tolerance analysis in a unified framework that is convenient for algorithmic processing and that is applicable not only in linear programming but for other linear systems as well. We survey the known results (pioneered by R.E. Wendell) and propose an improvement that is optimal in some sense (the resulting tolerances are maximal and they take into account proportionality). We apply our approach to several optimality invariancies: optimal basis, support set and optimal partition invariancy. Thus, the approach is useful not only for simplex method solvers, but for the interior points methods, too. We also discuss time complexity and show that it is NP-hard to determine the maximal tolerances.