On the stability of linear systems with an exact constraint set

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. Copyright Springer-Verlag 2006