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

Andrew Eberhard

RMIT

## Date:

Thu, 20/06/2019 - 10:00am to 11:00am

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

When restricted to a subspace, a nonsmooth function can be differentiable. This can be characterised for convex functions via a decomposition of the subdifferential giving rise to two subspaces: the $\mathcal{U}$, over which a special Lagrangian can be defined which has nice smooth properties and the $\mathcal{V}$ space, the orthogonal complement subspace of $\mathcal{U}$. Under the assumption of prox-regularity and the presence of a tilt stable local minimum one can 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. We will also consider the inverse problem. If a so-called “fast track” exists around a strict local minimum does this imply the existence of a tilt stable local minimum? We investigate conditions  under which this is so by studying the equivalence of the closely related notions of fast track and partial smoothness.  We will illustrate some of this with some graphical realisation of the BFGS method and various modifications of this method as applied to a nonsmooth Rosenbrock function which demonstrates that many questions remain unanswered as the algorithmic use of these ideas. (Joint work with Y. Luo and S. Liu)

Speaker's Bio: Andrew Eberhard did his PhD at Adelaide University under Prof. Charles Pearce and after graduating spend some time at UniSA before moving to RMIT in Melbourne in the 1990s. He has been an active member of the Australian mathematical and optimisation community for more than 20 years. He has served on the executives of ASOR, ANZIAM, as the deputy director of AMSI and served on the board of AMSI. Currently he is the co-chair of the AustMS special interest group Mathematics of Computation and Optimisation (MoCaO). His interests span numerous areas including nonsmooth and variational analysis, optimisation algorithms (both continuous and discrete), systems and control theory, operations research and other more theoretical aspect of optimisation theory.