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This paper addresses Lipschitzian stability issues that play an importantrole in both theoretical and numerical aspects of variational analysis,optimization, and their applications. We particularly concentrate on theso-called relaxed one-sided Lipschitz property of set-valued mappings withnegative Lipschitz constants. This property has been much less investigatedthan more conventional Lipschitzian behavior while being well recognized in avariety of applications. Recent work has revealed that set-valued mappingssatisfying the relaxed one-sided Lipschitz condition with negative Lipschitzconstant possess a localization property that is stronger than uniform metricregularity. The present paper complements this fact by providing acharacterization not only of one specific single point of a preimage, but ofentire preimages of such mappings. Developing a geometric approach, we derivean explicit formula to calculate preimages of relaxed one-sided Lipschitzmappings between finite-dimensional spaces and obtain a further specificationof this formula via extreme points of image sets.

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