On Fréchet differentiability of multifunctions

Abstract

The purpose of our article is to extend the classical notion of Frechet differentiability to multifunctions. To this end we define the notion of affinity for multifunctions and study the basic properties of affine multifunctions. Then using affine multifunctions as local approximations and the Hausdorff distance for defining an approximation mode, we introduce the notion of Frechet differentiability for multifunctions mapping points of a finite-dimensional normed space to compact convex subsets of another finite-dimensional normed space. We characterize Frechet differentiability of multifunctions through the differentiable properties of their support functions and discuss the relationship of our notion of differentiability with other ones.