Optimal-Value Functions

In the following nine chapters we study optimization problems whose formulations contain minimization and maximization operations in their description — optimization problems with a two-level structure. In many instances, these problems include optimal-value functions that are not necessarily differentiable and hence difficult to work with. In this chapter we highlight a number of important properties of optimal-value functions that derive from results by Clarke [C9], Gauvin-Dubeau [G4], Fiacco [F3], and Hogan [H12] on differentiable stability analysis for nonlinear programs. These results provide the basis for several computational techniques discussed presently.