Lipschitz-like property relative to a set and the generalized Mordukhovich criterion

TL;DR

This paper develops new theoretical conditions and a generalized criterion for the Lipschitz-like property of set-valued mappings relative to closed sets, using advanced tools like normal cones and projectional coderivatives.

Contribution

It introduces a verifiable generalized Mordukhovich criterion and characterizes Lipschitz-like properties relative to convex sets with new mathematical tools.

Findings

Complete characterization of Lipschitz property relative to convex sets.

Introduction of a projectional coderivative for set-valued mappings.

Representation of graphical modulus via projectional coderivative outer norm.

Abstract

In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.