On the stability of linear systems with an exact constraint set

Abstract

This paper deals with the stability of the intersection of a given set \( X\subset \mathbb{R}^{n}\)with the solution, \(F\subset \mathbb{R}^{n}\), of a given linear system whose coefficients can be arbitrarily perturbed. In the optimization context, the fixed constraint set X can be the solution set of the (possibly nonlinear) system formed by all the exact constraints (e.g., the sign constraints), a discrete subset of \(\mathbb{R}^{n}\) (as \( \mathbb{Z}^{n}\) or { 0,1} n, as it happens in integer or Boolean programming) as well as the intersection of both kind of sets. Conditions are given for the intersection \(F \cap X\) to remain nonempty (or empty) under sufficiently small perturbations of the data.