Convex Hulls of Algebraic Sets

Gouveia, João; Thomas, Rekha R.

doi:10.1007/978-1-4614-0769-0_5

Cited by 11 publications

(5 citation statements)

References 30 publications

(45 reference statements)

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“…We first recall the definition of a k-level polytope (see e.g., [GT12]): Definition 3. A polytope P is called k-level if every facet-defining linear function of P takes at most k different values on the vertices of the polytope.…”

Section: K-level Polytopes and Nonnegative Polynomial Interpolationmentioning

confidence: 99%

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Equivariant Semidefinite Lifts of Regular Polygons

2017

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Given a polytope P ⊂ 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 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 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 .

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Section: K-level Polytopes and Nonnegative Polynomial Interpolationmentioning

confidence: 99%

“…Any 2-level polytope has theta-rank one (see, e.g. [GT12]). One way to see this is to note that any sequence 0 = a 0 < a 1 of length 2 has nonnegative interpolation degree 2.…”

Section: Sketch Of Proof Let Q(x) =mentioning

confidence: 99%

Equivariant Semidefinite Lifts of Regular Polygons

2017

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“…Proving cπ T x + dt + e ≥ 0 is similar as in case 1. We only need to show that (x, t) satisfies the quadratic inequality in (42), which we prove by showing that…”

Section: Final Remarks and Future Workmentioning

confidence: 99%

“…Most of the known results in this area are limited to very specific sets [46,69,71] or to approximations of semi-algebraic sets through Semidefinite Programming (SDP) [37,51,60,61,62,63,64]. While some precise SDP representations of the convex hulls of semi-algebraic sets exist [42,44,45,68], these require the use of auxiliary variables. Such higher dimensional, extended, or lifted representations are extremely 1 INTRODUCTION 2 powerful.…”

Section: Introductionmentioning

confidence: 99%

Intersection cuts for nonlinear integer programming: convexification techniques for structured sets

2015

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We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. In particular, we give simple formulas for split cuts for essentially all convex sets described by a single quadratic inequality. We also give simple formulas for k-branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets.

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“…On the other hand, for a given basic closed semi-algebraic set, Lasserre developped in [Las09] a semi-definite relaxation method that was further investigated by Helton and Nie in [HN09] and [HN10]. Gouveia, Parrilo and Thomas constructed semi-definite relaxations of real algebraic sets in [GPT10] and [GT11]. Recently, Gouveia and Netzer further investigated exactness properties of these two methods, see [GN].…”

Section: Introductionmentioning

confidence: 99%

The Algebraic Boundary of SO(2)-Orbitopes

Sinn¹

2011

Preprint

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Let X ⊂ A 2r be a real curve embedded into an even-dimensional affine space. In the main result of this paper, we characterise when the r-th secant variety to X is an irreducible component of the algebraic boundary of the convex hull of the real points X(R) of X. This fact is then applied to 4-dimensional SO(2)-orbitopes and to the so called Barvinok-Novik orbitopes to study when they are basic closed as semialgebraic sets. In the case of 4-dimensional SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.

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Convex Hulls of Algebraic Sets

Cited by 11 publications

References 30 publications

Equivariant Semidefinite Lifts of Regular Polygons

Equivariant Semidefinite Lifts of Regular Polygons

Intersection cuts for nonlinear integer programming: convexification techniques for structured sets

The Algebraic Boundary of SO(2)-Orbitopes

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