Self-Scaled Barriers for Irreducible Symmetric Cones

Self{scaled barrier functions are fundamental objects in the theory of interior{point methods for linear optimization over symmetric cones, of which linear and semideenite 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 clas-siication 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 rst principles in the important special case of semideenite programming. We then generalise these arguments to irreducible symmetric cones by invoking results from the theory of Euclidean Jordan algebras.