Properties of the Moreau–Yosida regularization of a piecewise C2 convex function

In this paper we discuss second-order properties of the Moreau–Yosida regularization F of a piecewise twice continuously differentiable convex function f. We introduce a new constraint qualification in order to prove that the gradient of F is piecewise continuously differentiable. In addition, we discuss conditions, depending on the Hessians of the pieces, that guarantee positive definiteness of the generalized Jacobians of the gradient of F.