Preprint A284/2004
VU-Smoothness and Proximal Point Results for Some Nonconvex Functions

Claudia Sagastizabal | Mifflin, Robert

**Keywords: **
Proximal point | $\V\U$-decomposition | second-order derivatives.

This paper is concerned with a function $f$ having
primal-dual gradient structure at a point $\xb$ which satisfies a property called
strong transversality. The structure is related to $\V\U$-space decomposition,
depending on the subdifferential of $f$ at $\xb$. It is shown that there
exists a $C^2$ primal track leading to $\xb$ and a space decomposition mapping
that is $C^1$. As a result, there exists
a second order expansion of $f$ on the primal track,
an associated
subdifferential that is $C^1$ in a certain sense, and
a corresponding dual track.
For $\xb$ a minimizer, conditions on $f$ are given
to ensure that
for any point near $\xb$ its corresponding proximal point is on the primal track.