Variational Conditions and the Proto-differentiation of Partial Subgradient Mappings

Subgradient mappings have many roles in variational analysis, such as the formulation of optimality conditions and, in the special case of normal cone mappings, the statement and analysis of variational inequalities and related expressions. Central to the study of perturbations of solutions to systems of conditions in which subgradient mappings appear are concepts of generalized differentiation such as proto-differentiability, which is unhampered by a solution mapping’s potential multivaluedness. This paper extends the known examples of proto-differentiability by showing that a large and important class of “partial” subgradient mappings have this property. Until now the main examples have been the complete subgradient mappings associated with fully amenable functions. The extension, made difficult by the need to deal geometrically with projections of graphs, relies on the notion of a fully amenable function having additional variables that provide a “compatible parameterization.” The results are applied to the sensitivity analysis of generalized variational inequalities in which the underlying set need not be convex and can vary with the parameters.