On Lipschitzian properties of implicit multifunctions

TL;DR

This paper develops new sufficient conditions for the calmness and Aubin property of implicit multifunctions using directional limiting coderivatives and graphical derivatives, enhancing the analysis of their local behavior.

Contribution

It introduces novel criteria for calmness and Aubin property, including a new condition paralleling the implicit function paradigm, and provides formulas for computing directional limiting coderivatives.

Findings

New criteria for calmness and Aubin property

A formula for directional limiting coderivative of normal-cone maps

Examples illustrating the theoretical results

Abstract

The paper is devoted to the development of new sufficient conditions for the calmness and the Aubin property of implicit multifunctions. As the basic tool one employs the directional limiting coderivative which, together with the graphical derivative, enable us a fine analysis of the local behavior of the investigated multifunction along relevant directions. For verification of the calmness property, in addition, a new condition has been discovered which parallels the missing implicit function paradigm and permits us to replace the original multifunction by a substantially simpler one. Moreover, as an auxiliary tool, a handy formula for the computation of the directional limiting coderivative of the normal-cone map with a polyhedral set has been derived which perfectly matches the framework of [11]. All important statements are illustrated by examples.