Hyperbolicity cones are amenable

Bruno F. Lourenço, Vera Roshchina, James Saunderson

Research output: Contribution to journalArticleResearchpeer-review


Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or ‘nice’) which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.

Original languageEnglish
Pages (from-to)753–764
Number of pages12
JournalMathematical Programming
Issue numberSeries A
Publication statusPublished - 2024


  • Amenable cone
  • Facial structure
  • Hyperbolic polynomial
  • Hyperbolicity cone
  • Nice cone

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