Measuring the Stability. A Paradigmatic Problem in Optimization

Abstract This article aims to establish the fundamental notions on which stability analysis in optimization is based. It is a topic of singular importance in the analysis of any optimization problem, since the lack of stability can lead to serious problems in the application of the results. Our work here is of didactic nature and, consequently, it is limited to studying the basic criteria of qualitative and quantitative stability in Linear Programming. Formulas are provided that allow us to measure the level of each stability property by means of the so-called modulus. An important feature of our approach is that such moduli rely exclusively on the data of the nominal problem. The mathematical tools used in the paper are simple, consisting mainly in the analysis of lower/upper semicontinuity of set-valued mappings and, also, of their Lipschitz-type properties. Measuring the distance to ill-posed problems is another goal of this basic paper and some formulas to this purpose are shown. Most definitions are given for set-valued mappings between metric spaces, although the paper confines to Euclidean ones. The contents of the article could be incorporated into any introductory optimization course.