The 𝒰-Lagrangian of a convex function

Abstract

At a given point p ¯ \bar {p} , a convex function f f is differentiable in a certain subspace U \mathcal {U} (the subspace along which ∂ f ( p ¯ ) \partial f(\bar {p}) has 0-breadth). This property opens the way to defining a suitably restricted second derivative of f f at p ¯ \bar {p} . We do this via an intermediate function, convex on U \mathcal {U} . We call this function the U \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.