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

Roland Hildebrand (1)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques

Let K⊂Rn 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∗∈K0 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 ∂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 ν of logarithmically homogeneous self-concordant barriers F:K0→R on K in terms of the projective cross-ratios qk=(ek,x∗;yk,zk) . 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 ||⋅||∞ -norm in Rn and compute lower bounds on the barrier parameter for the power cone and the epigraph of the ||⋅||p -norm in R2 .

Mots clés: self-concordant barrier; barrier parameter

Domaines: Optimisation et contrôle [math.OC]

Roland Hildebrand. A lower bound on the barrier parameter of barriers for convex cones. Mathematical Programming, 2013, 142 (1-2), pp.311-329. ⟨10.1007/s10107-012-0576-1⟩. ⟨hal-00944376⟩