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 $\Aut(\Spsd)$.

      Item Type: Preprint
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: semidefinite programming, self-scaled barrier functionals, symmetric cones, interior-point methods
      Subjects: 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
      52-xx Convex and discrete geometry > 52Axx General convexity
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 28 Nov 2003
      Last Modified: 21 Apr 2010 11:13

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