  The optimal value function is one of the basic objects in the field of
mathematical optimization, as it allows the evaluation of the variations in the
cost/revenue generated while minimizing/maximizing a given function under some
constraints. In the context of stability/sensitivity analysis, a large number
of publications have been dedicated to the study of continuity and
differentiability properties of the optimal value function. The
differentiability aspect of works in the current literature has mostly been
limited to first order analysis, with focus on estimates of its directional
derivatives and subdifferentials, given that the function is typically
nonsmooth. With the progress made in the last two to three decades in major
subfields of optimization such as robust, minmax, semi-infinite and bilevel
optimization, and their connection to the optimal value function, there is a
crucial need for a second order analysis of the generalized differentiability
properties of this function. This type of analysis will enable the development
of robust solution algorithms, such as the Newton method. The main goal of this
paper is to provide results in this direction. In fact, we derive estimates of
the generalized Hessian for the optimal value function. Our results are based
on two handy tools from parametric optimization, namely the optimal solution
and Lagrange multiplier mappings, for which completely detailed estimates of
their generalized derivatives are either well-known or can easily be obtained.
