Polytopes of Minimum Positive Semidefinite Rank

Abstract

The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k\times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.

References in corpus (1)

• Exponential Lower Bounds for Polytopes in Combinatorial Optimization

Cited by in corpus (6)

• Lower bounds on nonnegative rank via nonnegative nuclear norms
• Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
• Worst-Case Results For Positive Semidefinite Rank
• Generalised probabilistic theories and conic extensions of polytopes
• The square root rank of the correlation polytope is exponential
• An upper bound for nonnegative rank