Upper Semicontinuity of the Feasible Set Mapping for Linear Inequality Systems

Abstract

In this paper we characterize the upper semicontinuity of the feasible set mapping at a consistent linear semi-infinite system (LSIS, in brief). In our context, no standard hypothesis is required in relation to the set indexing the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIS having the same index set, endowed with an extended metric to measure the size of the perturbations. We introduce the concept of reinforced system associated with our nominal system. Then, the upper semicontinuity property of the feasible set mapping at the nominal system is characterized looking at the feasible sets of both systems. The fact that this characterization depends only on the nominal system, not involving systems in a neighbourhood, is remarkable. We also provide a necessary and sufficient condition for the aimed property exclusively in terms of the coefficients of the system.