Stability Theory for Linear Inequality Systems II: Upper Semicontinuity of the Solution Set Mapping

Abstract

This paper deals with the upper semicontinuity of the solution set mapping for linear inequality systems, complementing a previous work on lower semicontinuity and related stability concepts. The main novelty of our approach is that we are not assuming any standard hypothesis about the set indexing the inequalities in the system. This set, possibly infinite, has no topological structure and, therefore, the functional dependence between the linear inequalities and their associated indices has no qualification at all. The space of consistent systems, over a fixed index set, is endowed with the uniform topology derived from the pseudometric of Chebyshev, which turns out to be a natural way to measure the size of the perturbations. In this context, we provide some necessary and some sufficient conditions for the upper semicontinuity of the feasible set map at a given system whose solution set is not necessarily bounded.