  In this paper we investigate the sensitivity analysis of parameterized
nonlinear variational inequalities of second kind in a Hilbert space. The
challenge of the present work is to take into account a perturbation on all the
data of the problem. This requires special adjustments in the definitions of
the generalized first- and second-order differentiations of the involved
operators and functions. Precisely, we extend the notions, introduced and
thoroughly studied by R.T. Rockafellar, of twice epi-differentiability and
proto-differentiability to the case of a parameterized lower semi-continuous
convex function and its subdifferential respectively. The link between these
two notions is tied to Attouch's theorem and to the new concept, introduced in
this paper, of convergent supporting hyperplanes. The previous tools allow us
to derive an exact formula of the proto-derivative of the generalized proximity
operator associated to a parameterized variational inequality, and deduce the
differentiability of the associated solution with respect to the parameter.
Furthermore, the derivative is shown to be the solution of a new variational
inequality involving semi- and twice epi-derivatives of the data. An
application is given to parameterized convex optimization problems involving
the sum of two convex functions (one of them being smooth). The case of smooth
convex optimization problems with inequality constraints is discussed in
details. This approach seems to be new in the literature and open several
perspectives towards theoretical and computational issues in nonlinear
optimization.
