The $\mathcal{U}$-Lagrangian of a convex function

At a given point ${\overline{p}}$, a convex function $f$ is differentiable in a certain subspace $\mathcal{U}$ (the subspace along which $\partial f({\overline{p}})$ has 0-breadth). This property opens the way to defining a suitably restricted second derivative of $f$ at ${\overline{p}}$. We do this via an intermediate function, convex on $\mathcal{U}$. We call this function the $\mathcal{U}$-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.