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Raphael Hauser (rah48damtp.cam.ac.uk) Abstract: Selfscaled barrier functions on selfscaled 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 longstep 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 selfscaled cones as the set of symmetric cones and develop a primaldual symmetric viewpoint on selfscaled 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 independent interest. We then show that any selfscaled barrier function decomposes in an essentially unique way into a direct sum of selfscaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of selfscaled barrier functions using the correspondence between symmetric cones and Euclidean Jordan algebras. Keywords: Selfscaled barrier functions, symmetric cones, decomposition of convex cones, Jordan algebras, universal barrier function, interiorpoint methods. Category 1: Linear, Cone and Semidefinite Programming (Other ) Category 2: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: Numerical Analysis Report DAMTP 2001/NA03, Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England CB3 9EW. March 2001. Download: [Compressed Postscript] Entry Submitted: 03/28/2001 Modify/Update this entry  
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