On strict proto-differentiability of set-valued mappings

TL;DR

This paper characterizes strict proto-differentiability of set-valued mappings via graphical differentiability, linking it to properties like metric regularity and deriving a second-order relation for prox-regular functions.

Contribution

It establishes a new equivalence between strict proto-differentiability and graphical strict differentiability, and introduces a second-order relation extending the trapezoidal rule.

Findings

Strict proto-differentiability is equivalent to graphical strict differentiability.

Under strict proto-differentiability, properties like strong metric regularity are equivalent.

A novel second-order relation for prox-regular functions extends numerical integration concepts.

Abstract

We will show that a multifunction is strictly proto-differentiable at a point of its graph if and only if it is graphically strictly differentiable, i.e., the graph of the multifunction locally coincides, up to a change of coordinates, with the graph of a single-valued mapping, which is strictly differentiable at the transformed reference point. This result allows point-based characterizations of strict proto-differentiability in terms of various generalized derivatives. Further we will prove that under strict proto-differentiability the properties of strong metric regularity, metric regularity and strong metric subregularity are equivalent. Finally, under strict proto-differentiability of the subgradient mapping, we provide a novel second-order relation between function values and subgradients for prox-regular functions which constitutes a nonsmooth extension of the trapezoidal rule of numerical integration.