An Extension of Attouch's Theorem and Its Application to Second-order Epi-differentiation of Convexly Composite Functions

In 1977, Hedy Attouch established that a sequence of (closed proper) convex functions epi-converges to a convex function if and only if the graphs of the subdifferentials converge (in the Mosco sense) to the subdifferential of the limiting function and (roughly speaking) there is a condition that fixes the constant of integration. We show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously difFerentiable mapping (in addition a constraint qualification is imposed). Using Attouch's Theorem, Rockafellar showed that second-order epi-differentiation of a convex function and proto-differentiability of the subdifferential set-valued mapping are equivalent, moreover the subdifferential of one-half the second-order epi-derivative is the proto-derivative of the subdifferential mapping; we will extend this result to the convexly composite setting.