Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones
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Author(s)
Saunderson, James F
Parrilo, Pablo A
Date Issued
August 2014
Journal
Mathematical Programming
Publisher
Springer Berlin Heidelberg
Citation
Saunderson, James, and Pablo A. Parrilo. “Polynomial-Sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones.” Mathematical Programming 153.2 (2015): 309–331.
Version
Author's final manuscript
Abstract
We give explicit polynomial-sized (in n and k) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree k in n variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in k,m, and n) of the hyperbolicity cones associated with kth directional derivatives of polynomials of the form p(x)=det(∑[superscript n][subscript i=1]A[subscript i]x[subscript i]) where the A[subscript i] are m×m symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1007/s10107-014-0804-y
