Self-Scaled Barrier Functions on Symmetric Cones and Their Classification

Self-scaled barrier functions on self-scaled cones were axiomatically introduced by Nesterov and Todd in 1994 as a tool for the construction of primal-dual long-step interior point algorithms. This paper provides firm foundations for these objects by exhibiting their symmetry properties, their close ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and their algebraic classification theory. In the first part we recall the characterization 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 independent interest. We then proceed to showing that any self-scaled barrier function decomposes, in an essentially unique way, into a direct sum 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.