  This paper aims at developing two versions of the generalized Newton method
to compute not merely arbitrary local minimizers of nonsmooth optimization
problems but just those, which possess an important stability property known as
tilt stability. We start with unconstrained minimization of continuously
differentiable cost functions having Lipschitzian gradients and suggest two
second-order algorithms of the Newton type: one involving coderivatives of
Lipschitzian gradient mappings, and the other based on graphical derivatives of
the latter. Then we proceed with the propagation of these algorithms to
minimization of extended-real-valued prox-regular functions, while covering in
this way problems of constrained optimization, by using Moreau envelops.
Employing advanced techniques of second-order variational analysis and
characterizations of tilt stability allows us to establish the solvability of
subproblems in both algorithms and to prove the Q-superlinear convergence of
their iterations.
