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Self-Scaled Barrier Functions: Decomposition and Classification

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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.

Contact: hauser@orie.cornell.edu