SELF { SCALED BARRIER FUNCTIONS ONSYMMETRIC CONES AND THEIR CLASSIFICATIONRaphael

نویسنده

  • Raphael A. Hauser
چکیده

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 rm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of deenition, and subsequently their decomposition into irreducible parts and algebraic classiication theory. In a rst 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 rst 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 self{scaled barrier function decomposes in an essentially unique way into a direct sum of self{scaled barriers deened on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classiication of self{scaled barrier functions using the correspondence between symmetric cones and Euclidean Jordan algebras. Abbreviated title. Self{scaled barrier functions.

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تاریخ انتشار 2002