Proximal Points are on the Fast Track

For a convex function, we consider a space decomposition that allows us to identify a subspace on which a Lagrangian related to the function appears to be smooth. We study a particular trajectory, that we call a fast track, on which a certain second-order expansion of the function can be obtained. We show how to obtain such fast tracks for a general class of convex functions having primal-dual gradient structure. Finally, we show that for a point near a minimizer its corresponding proximal point is on the fast track.