Selfscaled barrier functions on symmetric cones and their classification
Abstract
Selfscaled barrier functions on selfscaled cones were introduced through a set of axioms in 1994 by Y.E. Nesterov and M.J. Todd as a tool for the construction of longstep interior point algorithms. This paper provides firm foundation for these objects by exhibiting their symmetry properties, their intimate ties with the symmetry groups of their domains of definition, and subsequently their decomposition into irreducible parts and algebraic classification theory. In a first part we recall the characterisation of the family of selfscaled cones as the set of symmetric cones and develop a primaldual symmetric viewpoint on selfscaled barriers, results that were first discovered by the second author. We then show in a short, simple proof that any pointed, convex cone decomposes into a direct sum of irreducible components in a unique way, a result which can also be of independent interest. We then show that any selfscaled barrier function decomposes in an essentially unique way into a direct sum of selfscaled barriers defined on the irreducible components of the underlying symmetric cone. Finally, we present a complete algebraic classification of selfscaled barrier functions using the correspondence between symmetric cones and Euclidean Jordan algebras.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2001
 arXiv:
 arXiv:math/0103196
 Bibcode:
 2001math......3196H
 Keywords:

 Mathematics  Optimization and Control;
 90C25;
 90C60;
 52A41
 EPrint:
 17 pages