carath\’eodory number

A bound on the Carathéodory number

  • Masaru Ito
  • Bruno F. Lourenço
    • Categories Convex Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    The Carathéodory number k(K) of a pointed closed convex cone K is the minimum among all the k for which every element of K can be written as a nonnegative linear combination of at most k elements belonging to extreme rays. Carathéodory’s Theorem gives the bound k(K) <= dim (K). In this work we observe … Read more

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

  • Van Anh Truong
  • Levent Tunçel
    • Categories Linear, Cone and Semidefinite Programming Tags , , , , , , ,

    We study homogeneous convex cones. We first characterize the extreme rays of such cones in the context of their primal construction (due to Vinberg) and also in the context of their dual construction (due to Rothaus). Then, using these results, we prove that every homogeneous cone is facially exposed. We provide an alternative proof of … Read more