First- and Second-order Epi-differentiability in Nonlinear Programming

Problems are considered in which an objective function expressible as a max of finitely many C2 functions, or more generally as the composition of a piecewise linear-quadratic function with a C2 mapping, is minimized subject to finitely many C2 constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical firstand second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.