This paper addresses Lipschitzian stability issues, that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the so-called relaxed one-sided Lipschitz property of set-valued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional Lipschitzian behavior, while being well recognized in a variety of applications. Recent work has revealed that set-valued mappings satisfying the relaxed one-sided Lipschitz condition with negative Lipschitz constant possess a localization property, that is stronger than uniform metric regularity, but does not imply strong metric regularity. The present paper complements this fact by providing a characterization, not only of one specific single point of a preimage, but of entire preimages of such mappings. Developing a geometric approach, we derive an explicit formula to calculate preimages of relaxed one-sided Lipschitz mappings between finite-dimensional spaces and obtain a further specification of this formula via extreme points of image sets.

中文翻译：

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具有负 Lipschitz 常数的松弛单边 Lipschitz 映射原像的显式公式：一种几何方法

本文讨论了 Lipschitzian 稳定性问题，这些问题在变分分析、优化及其应用的理论和数值方面都起着重要作用。我们特别关注具有负 Lipschitz 常数的设置值映射的所谓松弛单边 Lipschitz 属性。与更传统的 Lipschitzian 行为相比，此属性的研究要少得多，但在各种应用中都得到了很好的认可。最近的工作表明，满足具有负 Lipschitz 常数的松弛单边 Lipschitz 条件的集合值映射具有定位特性，这比统一度量规则强，但并不意味着强度量规则。本论文通过提供一个特征来补充这一事实，不仅是原像的一个特定单点，而且是这种映射的整个原像。开发一种几何方法，我们推导出一个明确的公式来计算有限维空间之间松弛的单边 Lipschitz 映射的原像，并通过图像集的极值点获得该公式的进一步规范。