Accessible Requires Authentication Published by De Gruyter March 29, 2013

On the derivative cones of polyhedral cones

Raman Sanyal
From the journal

Abstract

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 derivative cones. In this note we give an affirmative answer for polyhedral cones by exhibiting an explicit spectrahedral representation for the first derivative cone. We also prove that higher polars do not have a determinantal representation which shows that the problem for general spectrahedral cones is considerably more difficult


This research was supported by a Miller Research Fellowship at UC Berkeley

Published Online: 2013-03-29
Published in Print: 2013-04

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