Theta Bodies for Polynomial Ideals

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

doi:10.1137/090746525

Cited by 121 publications

(190 citation statements)

References 29 publications

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“…Under the name theta bodies of V they have been studied by Gouveia, Parrilo, Thomas, and others (see [4] and [2,Chapter 7]). …”

Section: Letmentioning

confidence: 99%

“…Computing the convex hull of a set K ⊆ R n (that we assume to be basic closed semialgebraic) means to determine the linear moments of all probability measures on K for which these moments exist. By considering finite-dimensional relaxations of the K-moment problem, one obtains a nested hierarchy K(1) ⊇ K(2) ⊇ · · · of sets with explicit semidefinite representations that all contain K. Their closures K(d) = T H(d) have also been studied under the name theta bodies of K (see [4] and [2,Chapter 7]). When K is a compact semialgebraic set, the sets K(d) approximate conv(K) arbitrarily closely.…”

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confidence: 99%

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Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Scheiderer¹

2018

SIAM J. Appl. Algebra Geometry

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Abstract. We show that the closed convex hull of any one-dimensional semialgebraic subset of R n is a spectrahedral shadow, meaning that it can be written as a linear image of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve C and a compact semialgebraic subset K of its R-points, the preordering P(K) of all regular functions on C that are nonnegative on K is known to be finitely generated. Our main result, from which all others are derived, says that P(K) is stable, meaning that uniform degree bounds exist for weighted sum of squares representations of elements of P(K). We also extend this last result to the case where K is only virtually compact. The main technical tool for the proof of stability is the archimedean local-global principle. As a consequence of our results we show that every convex semialgebraic subset of R 2 is a spectrahedral shadow.

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“…Under the name theta bodies of V they have been studied by Gouveia, Parrilo, Thomas, and others (see [4] and [2,Chapter 7]). …”

Section: Letmentioning

confidence: 99%

mentioning

confidence: 99%

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Scheiderer¹

2018

SIAM J. Appl. Algebra Geometry

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“…While Las can be applied to semialgebraic sets, we restrict our discussion to its applications to polytopes contained in [0, 1] n . Gouveia, Parrilo and Thomas provided in [GPT10] an alternative description of the Las operator, where P I is described as the variety of an ideal intersected with the solutions to a system of polynomial inequalities. Our presentation of the operator is closer to that in [Lau03] than to Lasserre's original description.…”

Section: Preliminariesmentioning

confidence: 99%

Elementary polytopes with high lift-and-project ranks for strong positive semidefinite operators

Tunçel

2018

Discrete Optimization

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Abstract. We consider operators acting on convex subsets of the unit hypercube. These operators are used in constructing convex relaxations of combinatorial optimization problems presented as a 0,1 integer programming problem or a 0,1 polynomial optimization problem. Our focus is mostly on operators that, when expressed as a lift-and-project operator, involve the use of semidefiniteness constraints in the lifted space, including operators due to Lasserre and variants of the Sherali-Adams and Bienstock-Zuckerberg operators. We study the performance of these semidefinite-optimization-based lift-and-project operators on some elementary polytopes -hypercubes that are chipped (at least one vertex of the hypercube removed by intersection with a closed halfspace) or cropped (all 2 n vertices of the hypercube removed by intersection with 2 n closed halfspaces) to varying degrees of severity ρ. We prove bounds on ρ where these operators would perform badly on the aforementioned examples. We also show that the integrality gap of the chipped hypercube is invariant under the application of several lift-and-project operators of varying strengths.

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“…Then by Theorem 6, for any constraint equations (16)- (18) must hold. In particular, (16) and (17) imply that there can be only one violating assignment of the constraint g (x), and no assignment can be such that g (x) = 0.…”

Section: Problems With Linear Constraintsmentioning

confidence: 99%

On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

Kurpisz

Leppänen

Mastrolilli

2015

Automata, Languages, and Programming

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The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems.In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n − 1. These problems are the hardest for the Lasserre hierarchy in this sense.

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Theta Bodies for Polynomial Ideals

Cited by 121 publications

References 29 publications

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Elementary polytopes with high lift-and-project ranks for strong positive semidefinite operators

On the Hardest Problem Formulations for the $$0/1$$ Lasserre Hierarchy

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