Lipschitz upper semicontinuity of linear inequality systems under full perturbations

Abstract

The present paper is focused on the computation of the Lipschitz upper semicontinuity modulus of the feasible set mapping in the context of fully perturbed linear inequality systems, i.e. where all coefficients are allowed to be perturbed. The direct antecedent comes from the framework of right-hand side (RHS, for short) perturbations. The difference between both parametric contexts, full versus RHS perturbations, is emphasized. In particular, the polyhedral structure of the graph of the feasible set mapping in the latter framework enables us to apply classical results as those of Hoffman [On approximate solutions of systems of linear inequalities. J Res Natl Bur Stand. 1952;49:263–265] and Robinson [Some continuity properties of polyhedral multifunctions. Math Progr Study. 1981;14:206–214]. In contrast, the graph of the feasible set mapping under full perturbations is no longer polyhedral (not even convex). This fact requires ad hoc techniques to analyse the Lipschitz upper semicontinuity property and its corresponding modulus.