A lower bound on the barrier parameter of barriers for convex cones
成果类型：
Article
署名作者：
Hildebrand, Roland
署名单位：
Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
刊物名称：
MATHEMATICAL PROGRAMMING
ISSN/ISSBN：
0025-5610
DOI：
10.1007/s10107-012-0576-1
发表日期：
2013
页码：
311-329
关键词：
摘要：
Let K subset of R-n be a regular convex cone, let e(1), ... , e(n) is an element of partial derivative K be linearly independent points on the boundary of a compact affine section of the cone, and let x* is an element of K-0 be a point in the relative interior of this section. For k = 1, ... , n, let l(k) be the line through the points e(k) and x*, let y(k) be the intersection point of l(k) with partial derivative K opposite to e(k), and let z(k) be the intersection point of l(k) with the linear subspace spanned by all points e(l), l = 1, ... , n except e(k). We give a lower bound on the barrier parameter nu of logarithmically homogeneous self-concordant barriers F : K-0 -> R on K in terms of the projective cross-ratios q(k) = ( e(k), x*; y(k), z(k)). Previously known lower bounds by Nesterov and Nemirovski can be obtained from our result as a special case. As an application, we construct an optimal barrier for the epigraph of the ||.||(infinity)-norm in R-n and compute lower bounds on the barrier parameter for the power cone and the epigraph of the ||.||(infinity)-norm in R-2.