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 $$n$$n-dimensional polytope with $$v$$v vertices is at least $$(nv)^{\frac{1}{4}}$$(nv)14 improving on previous lower bounds. For polygons with $$v$$v vertices, we show that psd rank cannot exceed $$4 \left\lceil v/6 \right\rceil $$4v/6 which in turn shows that the psd rank of a $$p \times q$$p×q matrix of rank three is at most $$4\left\lceil \min \{p,q\}/6 \right\rceil $$4min{p,q}/6. In general, a nonnegative matrix of rank $${k+1 \atopwithdelims ()2}$$k+12 has psd rank at least $$k$$k and we pose the problem of deciding whether the psd rank is exactly $$k$$k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when $$k$$k is fixed.