A lower bound on the optimal self-concordance parameter of convex cones

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.

A lower bound on the barrier parameter of barriers for convex cones

Let K ⊂ R be a regular convex cone, let e1, . . . , en ∈ ∂K be linearly independent points on the boundary of a compact affine section of the cone, and let x∗ ∈ K be a point in the relative interior of this section. For k = 1, . . . , n, let lk be the line through the points ek and x ∗, let yk be the intersection point of lk with ∂K opposite to ek, and let zk be the intersection point of lk wit...

Some Results on Boundedness in Locally Convex Cones

The entropic barrier: a simple and optimal universal self-concordant barrier

We prove that the Fenchel dual of the log-Laplace transform of the uniform measure on a convex body in Rn is a (1 + o(1))n-self-concordant barrier. This gives the first construction of a universal barrier for convex bodies with optimal self-concordance parameter. The proof is based on basic geometry of log-concave distributions, and elementary duality in exponential families.

Primal-dual path-following algorithms for circular programming

Circular programming problems are a new class of convex optimization problems that include second-order cone programming problems as a special case. Alizadeh and Goldfarb [Math. Program. Ser. A 95 (2003) 3-51] introduced primal-dual path-following algorithms for solving second-order cone programming problems. In this paper, we generalize their work by using the machinery of Euclidean Jordan alg...