The -Lagrangian of a convex function

At a given point , a convex function is differentiable in a certain subspace (the subspace along which has 0-breadth). This property opens the way to defining a suitably restricted second derivative of at . We do this via an intermediate function, convex on . We call this function the -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.