# Self-scaled Barriers for Semidefinite Programming

Hauser, Raphael A. (2000) Self-scaled Barriers for Semidefinite Programming. [Preprint]

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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 $\Spsd$ of symmetric positive semidefinite matrices are homothetic transformation of the universal barrier functional. 2.\ All self-scaled barrier functionals for $\Spsd$ 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 $\Spsd$ are isotropic. As a consequence we find that a self-concordant barrier functional $H$ for $\Spsd$ is self-scaled if and only if $\Aut(\Spsd)$ acts as a group of translations on $H$, and that the closed subgroup of $\Aut(\Spsd)$ generated by the set of Hessians of a self-scaled barrier $H$ coincides with the orientation preserving part of $\Aut(\Spsd)$.