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

## TL;DR

This paper derives an explicit geometric formula for preimages of set-valued mappings with relaxed one-sided Lipschitz property and negative Lipschitz constants, enhancing understanding of their structure in finite-dimensional spaces.

## Contribution

It provides a novel explicit formula for preimages of relaxed one-sided Lipschitz mappings with negative constants, extending previous localization results.

## Key findings

- Explicit formula for preimages derived
- Formula refined using extreme points of image sets
- Applicable to finite-dimensional spaces

## 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.