On the derivative cones of polyhedral cones

Details On the derivative cones of polyhedral cones On the derivative cones of polyhedral cones

2011-05-15

arxiv.org/abs/1105.2924v2

Mathematics

Optimization and Control

5 pages, 2 figures; v2: minor changes suggested by referee, to appear in "Advances in Geometry"

Scientific paper

Hyperbolic polynomials elegantly encode a rich class of convex cones that includes polyhedral and spectrahedral cones. Hyperbolic polynomials are closed under taking polars and the corresponding cones, the derivative cones, yield relaxations for the associated optimization problem and exhibit interesting facial properties. While it is unknown if every hyperbolicity cone is a section of the positive semidefinite cone, it is natural to ask whether spectrahedral cones are closed under taking polars. In this note we give an affirmative answer for polyhedral cones by exhibiting an explicit spectrahedral representation for the first derivative cone. We also proof that higher polars do not have an determinantal representation which shows that the problem for general spectrahedral cones is considerably more difficult.

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