Worst-case results for positive semidefinite rank

Abstract

We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic $n$-dimensional polytope with $v$ vertices is at least $(nv)^{\frac{1}{4}}$ improving on previous lower bounds. For polygons with $v$ vertices, we show that psd rank cannot exceed $4 \left\lceil v/6 \right\rceil$ which in turn shows that the psd rank of a $p \times q$ matrix of rank three is at most $4\left\lceil \min \{p,q\}/6 \right\rceil$. In general, a nonnegative matrix of rank ${k+1 \atopwithdelims ()2}$ has psd rank at least $k$ and we pose the problem of deciding whether the psd rank is exactly $k$. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when $k$ is fixed.