Implicit Functions and Solution Mappings pp 197-250 | Cite as

# Regularity Properties Through Generalized Derivatives

## Abstract

In the wide-ranging generalizations we have been developing of the inverse function theorem and implicit function theorem, we have followed the idea that conclusions about a solution mapping, concerning the Aubin property, say, or the existence of a single-valued localization, can be drawn by confirming that some auxiliary solution mapping, obtained from a kind of approximation, has the property in question. In the classical framework, we can appeal to a condition like the invertibility of a Jacobian matrix and thus tie in with standard calculus. Now, though, we are far away in another world where even a concept of differentiability seems to be lacking. However, substitutes for classical differentiability can very well be introduced and put to work. In this chapter we show the way to that and explain numerous consequences.

First, graphical differentiation of a set-valued mapping is defined through the variational geometry of the mapping’s graph. A characterization of the Aubin property is derived and applied to the case of a solution mapping. Strong metric subregularity is characterized next. Applications are made to parameterized constraint systems and special features of solution mappings for variational inequalities. There is a review then of some other derivative concepts and the associated inverse function theorems of Clarke and Kummer. Finally, alternative results using coderivatives are described.