VU-Smoothness and Proximal Point Results for Some Nonconvex Functions
Claudia Sagastizabal | Mifflin, Robert
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.