Convex sets with semidefinite representation

Lasserre, Jean-Bernard

doi:10.1007/s10107-008-0222-0

Cited by 56 publications

(81 citation statements)

References 20 publications

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“…Then c d is the optimum of an explicit semidefinite program, and c d ↑ f * by the general results of [7]. Our results imply that we in fact have finite convergence, i.e., f * = c d for some d ∈ N which depends only on C and deg(f ), but not on f .…”

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confidence: 51%

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: 51%

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Scheiderer¹

2018

SIAM J. Appl. Algebra Geometry

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“…In the above, we have actually shown that a property called Schmüdgen's Bounded Degree Nonnegative Representation (S-BDNR) (see Helton and Nie [6]) holds, i.e., every affine polynomials ℓ T x − ℓ * nonnegative on T belongs to the preordering generated by the f ′ i s and h ′ j s with uniform degree bounds. This implies a weaker property called the Schmüdgen's Bounded Degree Representation (S-BDR) (see Lasserre [8]) holds, i.e., almost every affine polynomials ℓ T x − ℓ * positive on T belongs to the preordering generated by the f ′ i s and h ′ j s with uniform degree bounds. So Theorem 2 in [8] can be applied to show that the LMI (4.3) is a SDP representation of conv(T ).…”

Section: The Pdlh Conditionmentioning

confidence: 99%

“…This implies a weaker property called the Schmüdgen's Bounded Degree Representation (S-BDR) (see Lasserre [8]) holds, i.e., almost every affine polynomials ℓ T x − ℓ * positive on T belongs to the preordering generated by the f ′ i s and h ′ j s with uniform degree bounds. So Theorem 2 in [8] can be applied to show that the LMI (4.3) is a SDP representation of conv(T ). For the convenience of readers, we give the direct proof here.…”

Section: The Pdlh Conditionmentioning

confidence: 99%

“…When S is a basic closed semialgebraic set of the form {x ∈ R n : g 1 (x) ≥ 0, · · · , g m (x) ≥ 0}, there is recent work on the SDP representability of S and its convex hull. Parrilo [11] and Lasserre [8] independently proposed a natural construction of lifted LMIs using moments and sum of squares techniques with the aim of producing SDP representations. Parrilo [11] proved the construction gives an SDP representation in the two dimensional case when the boundary of S is a single rational planar curve of genus zero.…”

Section: Introductionmentioning

confidence: 99%

“…To obtain these conditions we give a new and different construction of SDP representations, which we combine with those discussed in [6,8,11]. The following are our main contributions.…”

Section: Introductionmentioning

confidence: 99%

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Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

Helton¹,

Nie²

2009

SIAM J. Optim.

105

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Abstract. A set S ⊆ R n is called to be Semidefinite (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to construct SDP representations.The contributions of this paper are: (i) For bounded SDP representable sets W 1 , · · · , Wm, we give an explicit construction of an SDP representation for conv(∪ m k=1 W k ). This provides a technique for building global SDP representations from the local ones.(ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient: the boundary ∂S is nonsingular and positively curved, while necessary is: ∂S has nonnegative curvature at each nonsingular point. In terms of defining polynomials for S, nonsingular boundary amounts to them having nonvanishing gradient at each point on ∂S and the curvature condition can be expressed as their strict versus nonstrict quasi-concavity of at those points on ∂S where they vanish. The gaps between them are ∂S having or not having singular points either of the gradient or of the curvature's positivity. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T , we find that the critical object is ∂cT , the maximum subset of ∂T contained in ∂conv(T ). We prove sufficient for SDP representability: ∂cT is nonsingular and positively curved, and necessary is: ∂cT has nonnegative curvature at nonsingular points. The gaps between our sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition, which meshes well with constructions, is also discussed.

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Semidefinite Optimization Applications

2011

Wiley Encyclopedia of Operations Research and Management Science

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Convex sets with semidefinite representation

Cited by 56 publications

References 20 publications

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Semidefinite Representation for Convex Hulls of Real Algebraic Curves

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

Semidefinite Optimization Applications

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