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Lipschitz upper semicontinuity of linear inequality systems under full perturbations

Jesús Camacho, María Josefa Cánovas, Helmut Gfrerer, Juan Parra

TL;DR

This work analyzes the Lipschitz upper semicontinuity modulus of the feasible set mapping for linear inequality systems under full perturbations, drawing a clear contrast with RHS perturbations where the graph is polyhedral. It provides a limit-based characterization of the modulus and shows that, unlike RHS perturbations, the full-perturbation case yields infinite Lipusc in the unbounded scenario, with finite values arising only in bounded or one-dimensional special cases. In the bounded case, Lipusc equals the supremum (and indeed the maximum) of the calmness moduli at feasible points, which can be computed via extreme points and the D-sets, culminating in a computable corollary expressed entirely in terms of nominal data. These results extend the existing RHS framework to the nonpolyhedral graph setting and offer practical formulas for stability assessment in fully perturbed linear inequality systems, with illustrations and directions for future work on continuity of calmness moduli.

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.

Lipschitz upper semicontinuity of linear inequality systems under full perturbations

TL;DR

This work analyzes the Lipschitz upper semicontinuity modulus of the feasible set mapping for linear inequality systems under full perturbations, drawing a clear contrast with RHS perturbations where the graph is polyhedral. It provides a limit-based characterization of the modulus and shows that, unlike RHS perturbations, the full-perturbation case yields infinite Lipusc in the unbounded scenario, with finite values arising only in bounded or one-dimensional special cases. In the bounded case, Lipusc equals the supremum (and indeed the maximum) of the calmness moduli at feasible points, which can be computed via extreme points and the D-sets, culminating in a computable corollary expressed entirely in terms of nominal data. These results extend the existing RHS framework to the nonpolyhedral graph setting and offer practical formulas for stability assessment in fully perturbed linear inequality systems, with illustrations and directions for future work on continuity of calmness moduli.

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.
Paper Structure (6 sections, 10 theorems, 68 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 68 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz upper semicontinuity of the feasible set mapping under full perturbations
  4. Unbounded feasible set
  5. Bounded feasible set
  6. Conclusions, illustrations and further research

Key Result

Theorem 1

Let $\mathcal{M}:Y\rightrightarrows X,$ with $Y$ being a normed space and $X$ being a reflexive Banach space. Assume that $\mathop{\mathrm{gph}}\nolimits\mathcal{M}$ is a nonempty convex set. For any $\overline{y}\in \mathop{\mathrm{dom}}\nolimits\mathcal{M}$ such that $\mathcal{M}(\overline{y})$ is

Figures (1)

  • Figure :

Theorems & Definitions (15)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Remark 1
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • Theorem 4
  • Remark 2
  • ...and 5 more