Self-Scaled Barriers for Semidefinite Programming

Raphael Hauser

We show a result that can be expressed in any of the following three equivalent ways: 1. All self-scaled barrier functionals for the cone $\Sigma_+$ of symmetric positive semidefinite matrices are homothetic transformation of the universal barrier functional. 2. All self-scaled barrier functionals for $\Sigma_+$ can be expressed in the form $X\mapsto -c_1\ln\det X +c_0$ for some constants $c_1>0,c_0\in\RN$. 3. All self-scaled barrier functionals for $\Sigma_+$ are isotropic. As a consequence we find that a self-concordant barrier functional $H$ for $\Sigma_+$ is self-scaled if and only if $\Aut(\Sigma_+)$ acts as a group of translations on $H$, and that the closed subgroup of $\Aut(\Sigma_+)$ generated by the set of Hessians of a self-scaled barrier $H$ coincides with the orientation preserving part of $\Aut(\Sigma_+)$.

Numerical Analysis Report DAMTP 2000/NA02, Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England CB3 9EW.