Lifts of Convex Sets and Cone Factorizations

Gouveia, João; Parrilo, Pablo A.; Thomas, Rekha R.

doi:10.1287/moor.1120.0575

Cited by 156 publications

(242 citation statements)

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“…Together with the result from [6], where it was noted that a sufficiently irregular convex hexagon has full extension complexity, Theorem 3.2 provides a full answer for Problem 1.3. Namely, the following result is true.…”

Section: Resultsmentioning

confidence: 72%

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An upper bound for nonnegative rank

Shitov

2014

Journal of Combinatorial Theory, Series A

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Abstract. We provide a nontrivial upper bound for the nonnegative rank of rank-three matrices, which allows us to prove that 6n 7 linear inequalities suffice to describe a convex n-gon up to a linear projection. PreliminariesConsider a convex polytope P ⊂ R n . An extension [5,8] of P is a polytope Q ⊂ R d such that P can be obtained from Q as an image under a linear projection from R d to R n . An extended formulation [8,10] of P is a description of Q by linear equations and linear inequalities (together with the projection). The size [8, 10] of the extended formulation is the number of facets of Q. The extension complexity [8, 10] of a polytope P is the smallest size of any extended formulation of P , that is, the minimal possible number of inequalities in the description of Q. The number of facets of Q can sometimes be significantly smaller [5] than that of P , and this phenomenon can be used to reduce the complexity of linear programming problems useful for numerous applications [3,5,10

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Section: Resultsmentioning

confidence: 72%

“…In [6] it was noted that a sufficiently irregular convex hexagon has full extension complexity, stating the positive answer for n = 6. For n ≥ 7, the problem has been open.…”

Section: Theorem 12 [10 Theorem 2]mentioning

confidence: 99%

An upper bound for nonnegative rank

Shitov

2014

Journal of Combinatorial Theory, Series A

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“…The concepts defined here are not used in the proofs of our theorems, but simply make some of the results more convenient to state. We refer the reader to [Yan91,GPT13] for more details. Let P ⊂ R d be a polytope.…”

Section: Lifts Of Polytopesmentioning

confidence: 99%

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

2016

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Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T . Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay( G, S)) with maximal cliques related to T . Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G (the characters of G).We apply our general result to two examples. First, in the case where G = Z n 2 , by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most n/2 , establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on ZN (when d divides N ). By constructing a particular chordal cover of the dth power of the N -cycle, we prove that any such function is a sum of squares of functions with at most 3d log(N/d) nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in R 2d with N vertices can be expressed as a projection of a section of the cone of positive semidefinite matrices of size 3d log(N/d). Putting N = d 2 gives a family of polytopes in R 2d with linear programming extension complexity Ω(d 2 ) and semidefinite programming extension complexity O(d log(d)). To the best of our knowledge, this is the first explicit family of polytopes (P d ) in increasing dimensions where xcPSD(P d ) = o(xcLP(P d )) (where xcPSD and xcLP are respectively the SDP and LP extension complexity).The authors are with the

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“…Work in the second group include Gouveia et al (2013), Ballerstein and Michaels (2014), Bärmann et al (2014), Buchanan and Butenko (2014), Godinho et al (2014), Lancia and Serafini (2014) and Leggieri et al (2014). Examples of work in the third group include Kaibel (2011), Fiorini et al (2011, 2012a, 2012b), Faenza et al (2012, Gillis and Glineur (2012) and Kaibel and Walter (2014).…”

Section: Introductionmentioning

confidence: 99%

Limits to the scope of applicability of extended formulations theory for LP models of combinatorial optimisation problems

Diaby

Karwan

2017

IJMOR

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Abstract:The purpose of this paper is to bring to attention and to make a contribution to the issue of defining/clarifying the scope of applicability of extended formulations (EFs) theory. Specifically, we show that EFs theory is not valid for relating the sizes of descriptions of polytopes when the sets of the descriptive variables for those polytopes are disjoint, and that new definitions of the notion of 'projection' upon which some of the recent extended formulations works [such as Kaibel (2011), Fiorini et al. (2011, 2012a, 2012b, Faenza et al. (2012), Gillis and Glineur (2012) and Kaibel and Walter (2014), for example] have been based can cause those works to over-reach in their conclusions.Keywords: extended formulations; combinatorial optimisation; computational complexity; linear programming.Reference to this paper should be made as follows: Diaby, M. and Karwan, M.H. (2017) 'Limits to the scope of applicability of extended formulations theory for LP models of combinatorial optimisation problems', Int.

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Lifts of Convex Sets and Cone Factorizations

Cited by 156 publications

References 32 publications

An upper bound for nonnegative rank

An upper bound for nonnegative rank

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

Limits to the scope of applicability of extended formulations theory for LP models of combinatorial optimisation problems

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