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

We make use of $\mathcal{VU}$-space decomposition theory to connect three minimization-oriented objects. These objects are $\mathcal{U}$-Lagrangians obtained from minimizing a function over $\mathcal{V}$-space, proximal points depending on minimization over $\mathbb{R}^n = \mathcal{U} \oplus \mathcal{V}$, 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 $\mathcal{U}$-Lagrangian. We also show that the function's proximal points are on a trajectory determined by certain $\mathcal{V}$-space minimizers.