Lipschitz upper semicontinuity of linear inequality systems under full perturbations

TL;DR

This paper investigates the Lipschitz upper semicontinuity of the feasible set mapping in fully perturbed linear inequality systems, extending previous RHS perturbation results to more complex, non-polyhedral cases.

Contribution

It introduces new techniques to analyze Lipschitz upper semicontinuity for fully perturbed systems where the feasible set graph is non-polyhedral, unlike in RHS perturbations.

Findings

Established bounds for the Lipschitz upper semicontinuity modulus.

Extended classical Hoffman and Robinson results to fully perturbed systems.

Provided ad hoc analytical methods for non-polyhedral feasible set graphs.

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 vs 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 [A. J. HOFFMAN, J. Res. Natl. Bur. Stand. 49 (1952), pp. 263--265] and Robinson [S. M. ROBINSON, Math. Progr. Study 14 (1981), pp. 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 analyze the Lipschitz upper semicontinuity property and its corresponding modulus.