Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

G Beer,J Parra,M A López,M J Cánovas

doi:10.1007/s10107-020-01543-9

Abstract

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in {mathbb {R}}^{n}. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of {mathbb {R}} ^{n+1}. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques.

Highlights

This paper is initially focussed on the Lipschitz behavior of the feasible set associated with a parametric family of linear inequality systems of the form:
Where x ∈ Rn is the vector of variables, CL Rn+1 is the parameter space of all nonempty closed subsets in Rn+1
When U is an infinite set, (1) is a linear semi-infinite inequality system. In this framework, perturbations fall on U and, so, obviously, two different systems, associated with different sets U1, U2 ∈ CL Rn+1, can have different cardinality. This setting includes as a particular case the parametric family of linear systems coming from linearizing convex inequalities of the form f (x) ≤ 0, f ∈ Γ, (2)

Summary

Introduction

This paper is initially focussed on the Lipschitz behavior of the feasible set associated with a parametric family of linear inequality systems of the form:. The first paper computes the Lipschitz modulus of the feasible set mapping in the context of linear systems with an arbitrarily fixed index set T of the form a′tx ≤ bt, t ∈ T ,. The results of [4] do not apply directly to our current setting unless some appropriate connection between both parameter spaces, CL Rn+1 and Rn+1 T , was established In relation to this point, we appeal to paper [5], which provides the motivation and background from the methodological point of view.

Preliminaries and first results

Indexation strategies and calmness of linear systems

On the stability of subdifferentials

Lipschitz modulus of F in the Hausdorff setting

Application to convex inequalities

The convex differentiable case