Self-Scaled Barrier Functions: Decomposition and Classi cation

The theory of self-scaled conic programming provides a uniied framework for the theories of linear programming, semideenite programming and convex quadratic programming with convex quadratic constraints. Nesterov and Todd's concept of self-scaled barrier functionals allows the exploitation of algebraic and geometric properties of symmetric cones in certain variants of the barrier method applied to self-scaled conic programming problems. In a rst part of this article we show that self-scaled barrier functionals can be decomposed into direct sums of self-scaled barrier functionals over the irreducible components of the underlying symmetric cone. Applying this decomposition theory in a second part, we give a complete classiication of the set of self-scaled barrier functionals that are invariant under the action of the orthogonal group of their conic domain of deenition (we call such functionals isotropic).