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

We make use of

-space decomposition theory to connect three minimization-oriented objects. These objects are

\mathcal{U} U

-Lagrangians obtained from minimizing a function over

\mathcal{V} V

-space, proximal points depending on minimization over

\mathbb{R}^n = \mathcal{U} \oplus \mathcal{V} R n = 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

\mathcal{U} U

-Lagrangian. We also show that the function's proximal points are on a trajectory determined by certain

\mathcal{V} V

-space minimizers.