7 problems

Let $A$ be a nonnegative matrix. Its PSD rank is the smallest integer $k$ for which there exist $k\times k$ real positive semidefinite matrices $B_1,\ldots,B_m$ and…

Let $M\in\mathbb R^{p\times q}_{\geq 0}$ have rank $k+1$ and satisfy … Let $P$ and $Q$ be the two polytopes corresponding to $M$ in the nested-polytope representation. Spectrahedro…

Rank-one factor abundance conjecture. Every such matrix has a psd factorization in which either at least $k$ matrices $A_i$ and $k-1$ matrices $B_j$ are rank one, or at least $k-1$…

Let $M$ be a matrix represented as a slack matrix $M=S_{P,Q}$ of nested polytopes $P\subseteq Q\subseteq\mathbb R^{r-1}$. A spectrahedral shadow $C$ of size $k$ between $P$ and $Q$…

Let $P$ be a psd-minimal $d$-polytope, and let its support matrix be the $0,1$ matrix with the same zero pattern as a slack matrix of $P$. Support-matrix rank conjecture. The suppo…

Let $P_1$ and $P_2$ be nonnegative matrices, and let $P_1\otimes P_2$ denote their tensor product. The positive semidefinite rank of a nonnegative matrix is the smallest $r$ for wh…

A polytope $P$ has positive semidefinite extension complexity equal to the smallest $r$ such that $P$ is the projection of an affine slice of the cone of $r$-dimensional positive s…