Semidefinite representation of convex sets

Helton, J. William; Nie, Jiawang

doi:10.1007/s10107-008-0240-y

Cited by 158 publications

(222 citation statements)

References 19 publications

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“…An interesting issue of further investigation is to provide concrete conditions on the concave polynomials g j 's, to ensure that the Lagrangian L f is s.o.s. The work in [5] provides some interesting results in this direction.…”

Section: Resultsmentioning

confidence: 97%

“…For instance, in Lasserre [12] one finds sets of sufficient conditions on the coefficients of a polynomial f to ensure it is s.o.s. Also, after the present paper was written, Helton and Nie [5] have provided several sufficient conditions for the Lagrangian L f to be s.o.s. In particular, if the Hessian −∇ 2 g j (X) can be written P j (X)P j (X) T for some (not necessarily square) matrix P j (X) (i.e.…”

Section: Sdr For Compact Convex Basic Semialgebraic Setsmentioning

confidence: 99%

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Convex sets with semidefinite representation

Lasserre

2008

Math. Program.

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Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed > 0, there is a convex set K such that co(K) ⊆ K ⊆ co(K) + B (where B is the unit ball of R n ), and K has an explicit SDr in terms of the gj's. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L f associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case.

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Section: Resultsmentioning

confidence: 97%

Section: Sdr For Compact Convex Basic Semialgebraic Setsmentioning

confidence: 99%

Convex sets with semidefinite representation

Lasserre

2008

Math. Program.

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“…In other respects, our assumptions are quite restrictive: we deal only with the convex case, and only consider perturbations to the objective, taking what is possibly just a first step towards a more general theory. Even the theoretical gain in generality in considering semi-algebraic sets is unclear, since, despite considerable interest and effort, an example of a semi-algebraic convex set that is not semidefinite representable remains undiscovered [14]. Nonetheless, this semi-algebraic approach is interesting: the main result is independent of the choice of presentation of the feasible region (a choice that may influence the corresponding genericity result in a complex fashion), the proof technique is novel in this context, the generic conclusion is stronger and more concrete (holding on a set that is dense and open rather than just full measure), and the sole assumption of semi-algebraicity is typically immediate to verify, due to the Tarski-Seidenberg Theorem.…”

Section: Introductionmentioning

confidence: 99%

Generic Optimality Conditions for Semialgebraic Convex Programs

Bolte¹,

Daniilidis

Lewis

2011

Mathematics of OR

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We consider linear optimization over a nonempty convex semialgebraic feasible region F . Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique "active" manifold, around which F is "partly smooth", and the second-order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F , and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F .

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“…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%

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

Helton¹,

Nie²

2009

SIAM J. Optim.

Self Cite

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 representation of convex sets

Cited by 158 publications

References 19 publications

Convex sets with semidefinite representation

Convex sets with semidefinite representation

Generic Optimality Conditions for Semialgebraic Convex Programs

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

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