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Self-scaled barrier functions on symmetric cones and their classification

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Raphael Hauser and Osman Guler

Self-scaled barrier functions on self-scaled cones were introduced
through a set of axioms in 1994 by Y.E. Nesterov and M.J. Todd
as a tool for the construction of long-step interior point algorithms.
This paper provides firm foundation for these objects by exhibiting
their symmetry properties, their intimate ties with the symmetry
groups of their domains of definition, and subsequently their
decomposition into irreducible parts and algebraic classification
theory. In a first part we recall the characterisation of the family of
self-scaled cones as the set of symmetric cones and develop a
primal-dual symmetric viewpoint on self-scaled barriers, results that
were first discovered by the second author. We then show in a short,
simple proof that any pointed, convex cone decomposes into a direct sum of
irreducible components in a unique way, a result which can also be of
self-scaled barriers defined on the irreducible components of the
underlying symmetric cone. Finally, we present a complete algebraic
classification of self-scaled barrier functions using the
correspondence between symmetric cones and Euclidean Jordan algebras.
Numerical Analysis Report DAMTP 2001/NA03,
Department of Applied Mathematics and
Theoretical Physics, Silver Street, Cambridge,
England CB3 9EW. March 2001.

Contact: rah48@damtp.cam.ac.uk