On Partial Smoothness, Tilt Stability and the $\mathcal{VU}$--Decomposition

TL;DR

This paper demonstrates that under prox-regularity and tilt stability, a $ ext{VU}$-decomposition yields a smooth manifold where the function behaves smoothly, advancing understanding of nonsmooth optimization structures.

Contribution

It establishes a link between $ ext{VU}$-decomposition and smooth manifold existence under prox-regularity and tilt stability, providing new insights into nonsmooth analysis.

Findings

Existence of a smooth manifold under given conditions

Function coincides with a smooth function on this manifold

Advances understanding of partial smoothness in optimization

Abstract

Under the assumption of prox-regularity and the presence of a tilt stable local minimum we are able to show that a $\mathcal{VU}$ like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function.