Self-Concordant Barriers for Cones Generated by Chebyshev Systems

Constructing self-concordant barriers for convex cones

In this paper we develop a technique for constructing self-concordant barriers for convex cones. We start from a simple proof for a variant of standard result [1] on transformation of a ν-self-concordant barrier for a set into a self-concordant barrier for its conic hull with parameter (3.08 √ ν + 3.57)2. Further, we develop a convenient composition theorem for constructing barriers directly fo...

Geometry of homogeneous convex cones, duality mapping, and optimal self-concordant barriers

Einstein-Hessian barriers on convex cones

On the interior of a regular convex cone K ⊂ R there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on K with parameter of order O(n), the universal barrier. This barrier is invariant with respect to the unimodular automorphism su...

Recursive Construction of Optimal Self-Concordant Barriers for Homogeneous Cones

In this paper, we give a recursive formula for optimal dual barrier functions on homogeneous cones. This is done in a way similar to the primal construction of Güler and Tunçel [1] by means of the dual Siegel cone construction of Rothaus [2]. We use invariance of the primal barrier function with respect to a transitive subgroup of automorphisms and the properties of the duality mapping, which i...

Characterization of the barrier parameter of homogeneous convex cones

We characterize the barrier parameter of the optimal self{concordant barriers for homogeneous cones. In particular, we prove that for homogeneous convex cones this parameter is the same as the rank of the corresponding Siegel domain. We also provide lower bounds on the barrier parameter in terms of the Carath eodory number of the cone. The bounds are tight for homogeneous self-dual cones.