A lower bound on the positive semidefinite rank of convex bodies


Type
Article
Change log
Authors
Din, Mohab Safey El 
Abstract

The positive semidefinite rank of a convex body C is the size of its smallest positive semidefinite formulation. We show that the positive semidefinite rank of any convex body C is at least logd where d is the smallest degree of a polynomial that vanishes on the boundary of the polar of C. This improves on the existing bound which relies on results from quantifier elimination. The proof relies on the B'ezout bound applied to the Karush-Kuhn-Tucker conditions of optimality. We discuss the connection with the algebraic degree of semidefinite programming and show that the bound is tight (up to constant factor) for random spectrahedra of suitable dimension.

Description
Keywords
math.OC, math.OC, cs.CC, cs.SC
Journal Title
SIAM Journal on Applied Mathematics
Conference Name
Journal ISSN
1095-712X
2470-6566
Volume Title
2
Publisher
Society for Industrial and Applied Mathematics