In this paper, we address the basic geometric question of when a given convex set is the
image under a linear map of an affine slice of a given closed convex cone. Such a …

Abstract Let M ∈ R^ p * q M∈ R p× q be a nonnegative matrix. The positive semidefinite
rank (psd rank) of M is the smallest integer k for which there exist positive semidefinite …

We use techniques from (tracial noncommutative) polynomial optimization to formulate
hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In …

This paper presents a selected tour through the theory and applications of lifts of convex
sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original …

The Theta rank of a finite point configuration V is the maximal degree necessary for a sum-of-
squares representation of a non-negative affine function on V. This is an important invariant …

An upper bound for nonnegative rank - ScienceDirect Skip to main contentSkip to article Elsevier
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2-level polytopes naturally appear in several areas of pure and applied mathematics,
including combinatorial optimization, polyhedral combinatorics, communication complexity …

The positive semidefinite (psd) rank of a polytope is the size of the smallest psd cone that
admits an affine slice that projects linearly onto the polytope. The psd rank of a d-polytope is …

Let A be an m× n matrix with nonnegative real entries. The psd rank of A is the smallest k for
which there exist two families (P 1,…, P m) and (Q 1,…, Q n) of positive semidefinite …

A (convex) polytope P is said to be 2-level if for each hyperplane H that supports a facet of P,
the vertices of P can be covered with H and exactly one other translate of H. The study of …