Seminar - Quoc Tran-Dinh

Structured smoothness in modern convex optimization

Abstract:

The importance of convex optimization techniques has dramatically increased in the last decade due to the rise of new theory for
structured sparsity and low-rankness, and successful statistical
learning models such as support vector machines. Convex optimization
formulations are now employed with great success in various subfields
of data sciences, including machine learning, compressive sensing,
medical imaging, geophysics, and bioinformatics.

However, the renewed popularity of convex optimization places convex
algorithms under tremendous pressure to accommodate increasingly
difficult nonlinear models and nonsmooth cost functions with ever
increasing data sizes. Overcoming these emerging challenges requires
nonconventional ways of exploiting useful yet hidden structures
within the underlying convex optimization models.

To this end, I will demonstrate how to exploit the classical notion of
smoothness in novel ways to develop fully rigorous methods for
fundamental convex optimization settings, from primal-dual framework
to composite convex minimization, and from proximal-path following
scheme to barrier smoothing technique. Some of these results play key
roles in convex optimization, such as unification and uncertainty
principles for augmented Lagrangian and decomposition methods, and have important computational implications such as solving convex programs on the positive semidefinite cone without any matrix decompositions.