Spectrahedral Shadows

Scheiderer, Claus

doi:10.1137/17m1118981

Cited by 38 publications

(30 citation statements)

References 30 publications

(40 reference statements)

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“…In 2009, Helton and Nie [5] conjectured that conversely every convex semialgebraic set is a spectrahedral shadow. This conjecture was recently disproved by the author [28]. In the present paper, however, we prove the existence of a semidefinite representation for the closed convex hull of any one-dimensional semialgebraic set in R n .…”

mentioning

confidence: 61%

“…Indeed, for every semialgebraic set K ⊆ R n of dimension at least two, there exists a polynomial map ϕ : R n → R N (for some N ≥ 1) such that the closed convex hull of ϕ(K) in R N has no semidefinite representation. This is proved in [28].…”

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confidence: 72%

“…Since the right bottom 3 × 3 submatrix of M is homogeneous, we can scale with Remark 5.3. It was already mentioned that the dimension hypothesis dim(K) ≤ 1 in Theorem 5.1 is essential, according to [28]. Similarly, this hypothesis is also essential for the stability result, Theorem 4.4, from which Theorem 5.1 was derived.…”

Section: Let Us Briefly Indicate How Theorem 41 Can Be Proved (mentioning

confidence: 92%

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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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confidence: 61%

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confidence: 72%

Section: Let Us Briefly Indicate How Theorem 41 Can Be Proved (mentioning

confidence: 92%

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Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Scheiderer¹

2018

SIAM J. Appl. Algebra Geometry

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“…Remark 3.3. Definition 3.2 expresses R f as a union of spectrahedral shadows [17,19]. To see this, fix a point x in V f .…”

Section: From Semidefinite To Quadratic Optimizationmentioning

confidence: 99%

The geometry of SDP-exactness in quadratic optimization

2019

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Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree.

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“…In particular, it is known that the conjecture is true in dimension 2 [Sch12]. The conjecture has been recently disproved by Scheidered, who showed that the cone of positive semidefinite forms cannot be expressed as a projection of spectrahedra, except in some particular cases [Sch16]. A comprehensive list of references can be found in this work.…”

Section: Introductionmentioning

confidence: 88%

The tropical analogue of the Helton–Nie conjecture is true

Allamigeon

Gaubert

Skomra

2019

Journal of Symbolic Computation

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Helton and Nie conjectured that every convex semialgebraic set over the field of real numbers can be written as the projection of a spectrahedron. Recently, Scheiderer disproved this conjecture [Sch16]. We show, however, that the following result, which may be thought of as a tropical analogue of this conjecture, is true: over a real closed nonarchimedean field of Puiseux series, the convex semialgebraic sets and the projections of spectrahedra have precisely the same images by the nonarchimedean valuation. The proof relies on game theory methods.

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Spectrahedral Shadows

Cited by 38 publications

References 30 publications

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

The geometry of SDP-exactness in quadratic optimization

The tropical analogue of the Helton–Nie conjecture is true

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