On the relation between U-hessians and second-order epiderivatives
Claudia Sagastizábal | Mifflin, Robert
Second-order epi-derivatives | $\U$-Hessians | epigraphical and pointwise convergence.
We consider second order objects for a convex function defined as the maximum of a finite number of $C^2$-functions. Variational analysis yields explicit formul\ae\/ for the second-order epi-derivatives of such max-functions. Another second-order object can be defined by means of a space decomposition that allows us to identify a subspace on which a Lagrangian related to a max-function is smooth. This decomposition yields an explicit expression for the so-called $\U$-Hessian of the function, defined as the Hessian of the related Lagrangian. We show that the second-order epi-derivative and the $\U$-Hessian are equivalent second-order objects. Thus, the $\U$-Lagrangian of a max-function captures the function's second-order epi-differential behavior with ordinary second derivatives.