Contingent derivatives of implicit (multi-) functions and stationary points

For an implicit multifunction Phi (p) defined by the generally nonsmooth eq uation F(x, p) = 0, contingent derivative formulas are derived, being simil ar to the formula Phi' = -F-x(-1) F-p in the standard implicit function the orem for smooth F and Phi. This will be applied to the projection X (p) = { x \ There Existsy: (x, y) is an element of Phi (p)} of the solution set Phi (p) of the system F(x, y, p) = 0 onto the x-space. In particular settings, X (p) may be interpreted as stationary solution sets. We discuss in detail the situation in which X(p) arises from the Karush-Kuhn-Tucker system of a nonlinear program.