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

Abstract

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