Polytopes of Minimum Positive Semidefinite Rank

@article{Gouveia2013PolytopesOM,
  title={Polytopes of Minimum Positive Semidefinite Rank},
  author={Jo{\~a}o Gouveia and Richard Z. Robinson and Rekha R. Thomas},
  journal={Discrete \& Computational Geometry},
  year={2013},
  volume={50},
  pages={679-699}
}
The positive semidefinite (psd) rank of a polytope is the smallest $$k$$k for which the cone of $$k \times k$$k×k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization… 

Complex psd-minimal polytopes in dimensions two and three

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

A representation-theoretic framework is presented to study equivariant PSD lifts of a certain class of symmetric polytopes known as orbitopes which respect the symmetries of the polytope.

The square root rank of the correlation polytope is exponential

This work shows that the square root rank of the slack matrix of the correlation polytope is exponential, and a way to lower bound the rank of certain matrices under arbitrary sign changes of the entries using properties of the roots of polynomials in number fields.

Positive semidefinite rank and nested spectrahedra

The geometry of this set of matrices is studied, and in small cases its boundary is described, and a conjecture for the description of this boundary is provided.

Positive semidefinite rank

The main mathematical properties of psd rank are surveyed, including its geometry, relationships with other rank notions, and computational and algorithmic aspects.

Equivariant Semidefinite Lifts of Regular Polygons

This paper shows that one can construct an equivariant psd lift of the regular 2^n-gon of size 2n-1, which is exponentially smaller than the psd Lift of the sum-of-squares hierarchy, and proves that the construction is essentially optimal.

Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

The power of lower bounds on positive semidefinite rank is characterized based on solely on the support of the matrix S, i.e., its zero/non-zero pattern.

Worst-case results for positive semidefinite rank

Using geometry and bounds on quantifier elimination, this decision can be made in polynomial time when k is fixed and it is proved that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv)^{\frac{1}{4}}$$(nv)14 improving on previous lower bounds.

References

SHOWING 1-10 OF 18 REFERENCES

Linear vs. semidefinite extended formulations: exponential separation and strong lower bounds

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if

Lifts of Convex Sets and Cone Factorizations

This paper addresses 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 and shows that the existence of a lift of a conveX set to a cone is equivalent to theexistence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual.

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP

Theta Bodies for Polynomial Ideals

A hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal is introduced and a geometric description of the first theta body for all ideals is given.

Extended Formulations for Polygons

It is proved that there exist n-gons whose vertices lie on an O(n)×O(n2) integer grid with extension complexity $\varOmega (\sqrt{n}/\ sqrt{\log n})$.

Smallest compact formulation for the permutahedron

It is shown how to obtain an extended formulation for this polytope from any sorting network, and it is shown that this is best possible (up to a multiplicative constant) since any extended formulation has at least $$varOmega (n \log n)$$Ω(nlogn) inequalities.

On the Shannon capacity of a graph

  • L. Lovász
  • Mathematics, Computer Science
    IEEE Trans. Inf. Theory
  • 1979
It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.

Constructing Extended Formulations from Reflection Relations

A framework of polyhedral relations is developed that generalizes inductive constructions of extended formulations via projections, and is particularly elaborate on the special case of reflection relations.