Preprint A146/2002
On the relation between U-hessians and second-order epiderivatives

Claudia Sagastizábal | Mifflin, Robert

**Keywords: **
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.