On Differential Properties of Multifunctions Defined Implicitly by Set-Valued Inclusions

TL;DR

This paper investigates the differential properties of solution mappings for set-valued inclusions using variational analysis, focusing on Lipschitzian behavior, graphical derivatives, and conditions for Aubin continuity.

Contribution

It introduces new conditions for Aubin continuity, characterizes the graphical derivative, and provides formulas for approximating derivatives of solution mappings in set-valued inclusions.

Findings

Established a condition for Aubin continuity via outer prederivative.

Derived formulas for inner and outer graphical derivatives.

Identified classes of convex solution mappings.

Abstract

In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the robust fulfilment of cone constraint systems, whose data are affected by a "crude knowledge" of uncertain elements, so they can not be casted in traditional generalized equations. The focus of this study in on the first-order behaviour of the solution mapping associated with a parameterized set-valued inclusion, starting with Lipschitzian properties and then considering its graphical derivative. In particular, a condition for the Aubin continuity of the solution mapping is established in terms of outer prederivative of the set-valued mapping defining the inclusion. A large class of parameterized set-valued inculsions is singled out, whose solution mapping turns out to be convex. Some relevant consequences on the graphical derivative are explored. In the absence of that, formulae for the inner and outer approximation of the graphical derivative are provided by means of prederivatives of the problem data. A representation useful to calculate the coderivative of the solution mapping is also obtained via the subdifferential of a merit function.