Proto-Differentiability of Set-Valued Mappings and its Applications in Optimization

Abstract

Proto-differentiability of a set-valued mapping (multifunction) G from one Euclidean space to another is defined in terms of graphical convergence of associated difference quotient multifunctions. The nature and consequences of the property are investigated in considerable detail to provide a basis for applications. Applications are demonstrated for the theory of optimization by verifying the proto-differentiability of some of the most important multifunctions in that theory, specifically multifunctions giving the set of solutions to a parameterized system of constraints or to a parameterized variational inequality or collection of optimality conditions. The fact that such multifunctions are actually differentiable in this a generalized sense has not previously been detected.