Explicit Formula for Preimages of Relaxed One-Sided Lipschitz Mappings with Negative Lipschitz Constants

Abstract: 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. 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.