Condition Number Minimization in Euclidean Jordan Algebras
Alberto Seeger · SIAM Journal on Optimization · 2022
Let $b$ be a nonnegative vector in a Euclidean Jordan algebra $\mathbb{E}$. Its condition number $\kappa(b)$ is assumed to be high, possibly infinite. The condition number of a nonnegative vector is defined as the ratio between the largest and the smallest eigenvalue. We wish to perturb $b$ in such a way as to diminish its condition number as much as possible. The problem at hand is that of minimizing $\kappa(b+x)$ with respect to a perturbation vector $x$ taken from a given subset of $\mathbb{E}$. In spite of being nonsmooth and nonconvex, such an optimization problem can be handled with the machinery of Euclidean Jordan algebras and, in particular, with the theory of spectral functions.