The U-Lagrangian of a convex function

At a given point (p) over bar, a convex function f is differentiable in a c ertain subspace U (the subspace along which partial derivative f((p) over b ar) has 0-breadth). This property opens the way to defining a suitably rest ricted second derivative of f at (p) over bar. We do this via an intermedia te function, convex on U. We call this function the U-Lagrangian; it coinci des with the ordinary Lagrangian in composite cases: exact penalty, semidef inite programming. Also, we use this new theory to design a conceptual patt ern for superlinearly convergent minimization algorithms. Finally, we estab lish a connection with the Moreau-Yosida regularization.