Polytopes of Minimum Positive Semidefinite Rank

Gouveia, João; Robinson, Richard Z.; Thomas, Rekha R.

doi:10.1007/s00454-013-9533-x

Cited by 36 publications

(45 citation statements)

References 18 publications

(45 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A characterization of psd-minimal polytopes in small dimensions was obtained in [GRT13] and, in particular, the following relation was shown. In [GRT13] an example of a psd-minimal polytope that is not 2-level is given, showing that the condition above is sufficient but not necessary.…”

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

“…In [GRT13] an example of a psd-minimal polytope that is not 2-level is given, showing that the condition above is sufficient but not necessary. The main result of this section is that the situation is much better for base configurations.…”

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

“…The psd rank of a polytope P is the smallest 'size' of a spectrahedron that linearly projects to P . The psd rank was studied in [GPT13,GRT13] and it was shown that the psd rank rank Psd (P ) is at least dim P + 1. Part (v) shows that the 2-level matroids are exactly those matroids for which the psd rank of the base polytope P M = conv(V M ) is minimal.…”

Section: Theorem 11 ([Gpt10 Thm 42])mentioning

confidence: 99%

“…Let us define V Psd k as the class of point configurations V in convex position such that conv(V ) has a psd-lift of size ≤ k. In [GRT13] it was shown that for a d-dimensional polytope P the psd rank is always ≥ d + 1. A polytope P is called psd-minimal if rank Psd (P ) = dim P + 1.…”

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

“…Since V Psd min is face-hereditary, it is sufficient to show that the excluded minors M (K 4 ), W 3 , Q 6 , and P 6 are not psd-minimal. In order to do so, we need to recall the connection to slack matrices and Hadamard square roots developed in [GRT13]. For a more coherent picture of the relations in particular to cone factorizations we refer to the papers [GPT13,GRT13].…”

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

See 4 more Smart Citations

Theta rank, levelness, and matroid minors

Grande

Sanyal

2017

Journal of Combinatorial Theory, Series B

View full text Add to dashboard Cite

Abstract. 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 for polynomial optimization that is in general hard to determine. We study the Theta rank and levelness, a related discrete-geometric invariant, for matroid base configurations. It is shown that the class of matroids with bounded Theta rank or levelness is closed under taking minors. This allows for a characterization of matroids with bounded Theta rank or levelness in terms of forbidden minors. We give the complete (finite) list of excluded minors for Theta-1 matroids which generalizes the well-known series-parallel graphs. Moreover, the class of Theta-1 matroids can be characterized in terms of the degree of generation of the vanishing ideal and in terms of the psd rank for the associated matroid base polytope. We further give a finite list of excluded minors for k-level graphs and matroids and we investigate the graphs of Theta rank 2.

show abstract

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

Section: Theorem 11 ([Gpt10 Thm 42])mentioning

confidence: 99%

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

Section: Proof From Proposition 42 We Already Know Thatmentioning

confidence: 99%

See 3 more Smart Citations