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.

中文翻译：

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具有 Hausdorff 度量的线性和凸不等式系统的 Lipschitz 模数

本文分析了 $${\mathbb {R}}^{n}$$ 中与线性和凸不等式系统相关的可行集映射的 Lipschitz 行为。首先，我们处理由相应系数向量集标识的线性（有限/半无限）系统的参数空间，假设它们是 $${\mathbb {R}} ^{n+ 1}$$。在这个框架中，扰动的大小是通过（扩展的）豪斯多夫距离来测量的。文献中广泛研究的直接前因来自考虑具有固定索引集 T 的所有线性系统的参数空间，其中使用切比雪夫（扩展）距离来测量扰动。在目前的工作中，我们提出了一种适当的索引策略，它允许我们在两个参数上下文中建立可行集映射的 Lipschitz 模的相等性，并从 Chebyshev 设置中的现有结果中受益，以便将它们转移到 Hausdorff 设置. 在第二阶段，通过线性化技术直接扰动线性系统的系数向量集的可能性导致对凸系统的 Lipschitz 行为的新贡献。