First- and second-order epi-differentiability in nonlinear programming

Abstract

Problems are considered in which an objective function expressible as a max of finitely many C 2 {C^2} functions, or more generally as the composition of a piecewise linear-quadratic function with a C 2 {C^2} mapping, is minimized subject to finitely many C 2 {C^2} 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 first-and second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.