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Self-scaled barriers for irreducible symmetric cones

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Raphael Hauser and Yongdo Lim

Self-scaled barrier functions are fundamental objects
in the theory of interior-point methods for linear optimization
over symmetric cones, of which linear and semidefinite programming
are special cases. We are classifying all self-scaled barriers
over irreducible symmetric cones and show that these functions are
merely homothetic transformations of the universal barrier
function. Together with a decomposition theorem for self-scaled
barriers this concludes the algebraic classification theory of
these functions. After introducing the reader to the concepts
relevant to the problem and tracing the history of the subject, we
start by deriving our result from first principles in the
important special case of semidefinite programming. We then
generalise these arguments to irreducible symmetric cones by
invoking results from the theory of Euclidean Jordan algebras.
Numerical Analysis Report DAMTP 2001/NA04,
Department of Applied Mathematics and Theoretical
Physics, Silver Street, Cambridge, England CB3 9EW.
April 2001.

Contact: rah48@damtp.cam.ac.uk