Proto-derivatives and the Geometry of Solution Mappings in Nonlinear Programming

We quantify the sensitivity of KKT pairs associated with a parameterized family of nonlinear programming problems. Our approach involves proto-derivatives, which are generalized derivatives appropriate even in cases when the KKT pairs are not unique; we investigate what the theory of such derivatives yields in the special case when the KKT pairs are unique (locally). We demonstrate that the graph of the KKT multifunction is just a reoriented graph of a Lipschitz mapping, and use proto-differentiability to show that the graph of the KKT multifunction actually has the stronger property of being a reorientation of the graph of a B-differentiable mapping. Our results indicate that protoderivatives provide the same kind of information for possibly set-valued mappings (like the KKT multifunction) that B-derivatives provide for single-valued mappings.