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We aim to construct the canonical barrier with parameter 3 for a quasi-polyhedral 3-dimensional cone defined by countably many linear inequalities, namely the closed convex conic hull of the set of integer points \((1,n,n^2) \in \mathbb R^3\) . The canonical barrier on 3-dimensional cones can in principle be computed by means of a general theory, demanding, in particular, the integration of a matrix-valued partial differential equation, the frame equation. For the particular cone in question we solve the arising problem of choosing the initial condition of this PDE. The construction is essentially using the non-trivial automorphisms and self-duality of the cone.

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Construction of a Self-concordant Barrier for a Quasi-Polyhedral Cone with Infinitely Many Faces

  • 会议论文

  • 发表日期：

    2025年

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  • Rahaf Habib,
  • Roland Hildebrand

摘要

We aim to construct the canonical barrier with parameter 3 for a quasi-polyhedral 3-dimensional cone defined by countably many linear inequalities, namely the closed convex conic hull of the set of integer points \((1,n,n^2) \in \mathbb R^3\) . The canonical barrier on 3-dimensional cones can in principle be computed by means of a general theory, demanding, in particular, the integration of a matrix-valued partial differential equation, the frame equation. For the particular cone in question we solve the arising problem of choosing the initial condition of this PDE. The construction is essentially using the non-trivial automorphisms and self-duality of the cone.