Contingent Derivatives of Implicit (Multi-) Functions and Stationary Points

For an implicit multifunction (p) defined by the generally nonsmooth equation F(x,p)=0, contingent derivative formulas are derived, being similar to the formula =–F _x ^–1 F _p in the standard implicit function theorem for smooth F and . This will be applied to the projection X(p)={xy: (x,y)(p)} of the solution set (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.