Lipschitz modulus of linear and convex systems with the Hausdorff metric

Abstract

This paper analyzes the Lipschitz behavior of the feasible set in two parametric settings, associated with linear and convex systems in 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 R^(n+1). In this framework, where the Hausdorff distance is used to measure the size of perturbations, an explicit formula for computing the Lipschitz modulus of the feasible set mapping is provided. As direct antecedent, we appeal to its counterpart in the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (pseudo) distance was considered to measure the perturbations. Indeed, the stability (and, particularly, Lipschitz properties) of linear systems in the Chebyshev framework has been widely analyzed in the literature. Here, through an appropriate indexation strategy, we take advantage of previous results to derive the new ones in the Hausdorff setting. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system allows us to provide new contributions on the Lipschitz behavior of convex systems via linearization techniques.