"Smoothing techniques on Jordan algebras"
Michel Baes (CORE / UCL)
Abstract:
A spectral function on a formally real Jordan algebra is a real-valued function
that depends only on the eigenvalues of its argument. One convenient way to
create them is to start from a function $f$ from $mathbb{R}^r$ to $mathbb{R}$
that is symmetric in the components of its argument, and to define the function
$F(u):=f(lambda(u))$ where $lambda(u)$ is the vector of eigenvalues of $u$. A
particular example of this construction is given by functions of symmetric
matrices that only depend on the eigenvalues of its argument. We positively
answer an open question proposed in the PhD thesis of H. Sendov : "is it
possible to compute the hessian of spectral functions on Jordan algebras?" We
then exploit this result to investigate how the power smoothing techniques of
Yu. Nesterov can in some instances be extended to Jordan algebras. In
particular, we propose a new algorithm to minimize a sum of Euclidean norms and
we perform its complexity analysis.
Other applications of our result include a new procedure for optimizing the
largest eigenvalue of symmetric matrices.
Two OPTEC professors have been awarded three "Gouden Krijtjes", the yearly teaching awards given by the organization of engineering students (vtk). Prof. Lombaert was awarded the prize for the best course in civil engineering, and Prof. Diehl the prizes for the best professor and the best course in mathematical engineering (where he teaches numerical optimization). They received these awards at the yearly "proffentap" where experienced students taught them how to draft beer professionally.
