Self-Scaled Barrier Functions: Decomposition and Classification

Raphael Hauser

The theory of self-scaled conic programming provides a unified framework for the theories of linear programming, semidefinite 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 first 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 classification of the set of self-scaled barrier functionals that are invariant under the action of the orthogonal group of their conic domain of definition (we call such functionals isotropic).

DAMTP-Report NA1999/13; Department of Applied Mathematics and Theoretical Physics, Silver Street, University of Cambridge, Cambridge CB3 9EW, England; October 1999.