On the equivalence of parametric contexts for linear inequality systems

In this paper we analyze the connections among different parametric settings in which the stability theory for linear inequality systems may be developed. Our discussion is focussed on the existence, or not, of an index set (possibly infinite). For some stability approaches it is not convenient to have a fixed set indexing the constraints. This is the case, for example, of discretization techniques viewed as approximation strategies (i.e., discretization regarded as data perturbation). The absence of a fixed index set is also a key point in the stability analysis of parametrized convex systems via standard linearization. In other frameworks the index set is very useful, for example if the constraints are perturbed one by one, even to measure the global perturbation size. This paper shows to what extent an index set may be introduced or removed in relation to stability.