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