Stability of the feasible set for linear inequality systems: A carrier index set approach

Abstract

In this paper we consider a parametrized family of linear inequality systems whose coefficients depend continuously on a parameter ranging in an arbitrary metric space. We analyze the lower semicontinuity (lsc) of the feasible set mapping in terms of the so-called carrier index set, consisting of those indices whose associated inequalities are satisfied as equalities at every feasible point. This concept, which leads to a weakened Slater condition, allows us to characterize the lsc of the feasible set mapping in terms of certain convex combinations of the coefficient vectors associated with the carrier indices. This property entails the lsc of the carrier feasible set mapping, assigning to each parameter the affine hull of the feasible set, which is also analyzed in this paper. The last section is concerned with the semi-infinite case.