Convex Analysis on a Class of Matrix Support Functionals
Tim Hoheisel, Ph.D.
In the first part of the talk, we introduce some key objects in convex analysis in finite-dimensional Euclidean space like, e.g., the relative interior and the convex hull of a convex set as well as the convex subdifferential and the Fenchel conjugate for extended real-valued functions. In particular, we will discuss Hörmander's Theorem which fully characterizes all proper, sublinear and lower semicontinuous functionals as support functions.
In the second part of the talk, a new class of matrix support functionals is presented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, and the nuclear norm. The support function is based on a set which is the graph of the product of a matrix with its transpose, intersected with an affine manifold. Some very recent results on the representation of the closed convex hull of said set are presented. This new representation allows the ready computation of a range of important related geometric objects whose formulations were previously unavailable, and which opens the door for various future applications in machine learning and related fields.
All are welcome to attend!