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Self-Scaled Barriers for Semidefinite Programming

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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.

Contact: rah48@damtp.cam.ac.uk