Relating U-Lagrangians to Second-Order Epi-Derivatives and Proximal Tracks

We make use of VU-space decomposition theory to connect three minimization-oriented objects. These objects are U-Lagrangians obtained from minimizing a function over V-space, proximal points depending on minimization over Rn = U + V, and epi-derivatives determined by lower limits associated with epigraphs. We relate second-order epi-derivatives of a function to the Hessian of its associated U -Lagrangian. We also show that the function's proximal points are on a trajectory determined by certain V-space minimizers.