dc.contributor.author Fawzi, Hamza en dc.contributor.author Saunderson, J en dc.contributor.author Parrilo, PA en dc.date.accessioned 2017-08-15T10:39:38Z dc.date.available 2017-08-15T10:39:38Z dc.date.issued 2017-05 en dc.identifier.issn 0364-765X dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/266381 dc.description.abstract Given a polytope P in $\mathbb{R}^n$, we say that P has a positive semidefinite lift (psd lift) of size d if one can express P as the linear projection of an affine slice of the positive semidefinite cone $\mathbf{S}^d_+$. If a polytope P has symmetry, we can consider equivariant psd lifts, i.e. those psd lifts that respect the symmetry of P. One of the simplest families of polytopes with interesting symmetries are regular polygons in the plane, which have played an important role in the study of linear programming lifts (or extended formulations). In this paper we study equivariant psd lifts of regular polygons. We first show that the standard Lasserre/sum-of-squares hierarchy for the regular N-gon requires exactly ceil(N/4) iterations and thus yields an equivariant psd lift of size linear in N. In contrast we show 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. Our construction relies on finding a sparse sum-of-squares certificate for the facet-defining inequalities of the regular 2^n-gon, i.e., one that only uses a small (logarithmic) number of monomials. Since any equivariant LP lift of the regular 2^n-gon must have size 2^n, this gives the first example of a polytope with an exponential gap between sizes of equivariant LP lifts and equivariant psd lifts. Finally we prove that our construction is essentially optimal by showing that any equivariant psd lift of the regular N-gon must have size at least logarithmic in N. dc.description.sponsorship This work was supported by the Air Force Office of Scientific Research [Grants FA9550-11-1-0305 and FA9550-12-1-0287]. dc.language.iso en en dc.publisher Institute for Operations Research and the Management Sciences dc.subject math.OC en dc.subject math.OC en dc.subject cs.CC en dc.subject cs.CG en dc.subject math.CO en dc.title Equivariant semidefinite lifts of regular polygons en dc.type Article prism.endingPage 494 prism.issueIdentifier 2 en prism.publicationDate 2017 en prism.publicationName Mathematics of Operations Research en prism.startingPage 472 prism.volume 42 en dc.identifier.doi 10.17863/CAM.9715 dcterms.dateAccepted 2016-06-14 en rioxxterms.versionofrecord 10.1287/moor.2016.0813 en rioxxterms.version AM en rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved en rioxxterms.licenseref.startdate 2017-05 en dc.contributor.orcid Fawzi, Hamza [0000-0001-6026-4102] dc.identifier.eissn 1526-5471 rioxxterms.type Journal Article/Review en cam.issuedOnline 2016-11-16 en dc.identifier.url https://pubsonline.informs.org/doi/10.1287/moor.2016.0813 en rioxxterms.freetoread.startdate 2018-08-24
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