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 ′ = −Fx−1Fp in the standard implicit function theorem for smooth F and . This will be applied to the projection X(p) = {x | ∃y: (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.