Pure states, nonnegative polynomials and sums of squares

Burgdorf, Sabine; Scheiderer, Claus; Schweighofer, Markus

doi:10.4171/cmh/250

Cited by 17 publications

(16 citation statements)

References 15 publications

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“…A slightly weaker version of this result was already proved in [3] Theorem 6.5. We give a new proof which is considerably shorter than the proof in [3]. Proof.…”

Section: Archimedean Local-global Principle For Semiringsmentioning

confidence: 95%

A Positivstellensatz for forms on the positive orthant

Scheiderer

Tan

2017

Arch. Math.

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Let p be a nonconstant form in R[x 1 , . . . , x n ] with p(1, . . . , 1) > 0. If p m has strictly positive coefficients for some integer m ≥ 1, we show that p m has strictly positive coefficients for all sufficiently large m.More generally, for any such p, and any form q that is strictly positive on (R + ) n \ {0}, we show that the form p m q has strictly positive coefficients for all sufficiently large m. This result can be considered as a strict Positivstellensatz for forms relative to (R + ) n \ {0}. We give two proofs, one based on results of Handelman, the other on techniques from real algebra.2010 Mathematics Subject Classification. Primary 12D99; secondary 14P99, 26C99 .

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“…A slightly weaker version of this result was already proved in [3] Theorem 6.5. We give a new proof which is considerably shorter than the proof in [3]. Proof.…”

Section: Archimedean Local-global Principle For Semiringsmentioning

confidence: 95%

A Positivstellensatz for forms on the positive orthant

Scheiderer

Tan

2017

Arch. Math.

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“…The concept of pure states stems from functional analysis and has recently entered commutative algebra [BSS]. Pure states on rings often just correspond to ring homomorphisms into the real numbers and therefore can be seen as real points of a variety [BSS,Propositions 4.4. and 4.16]. In this article, the variety concerned is just affine space.…”

Section: Systems Of Polynomial Inequalities and Polynomial Optimizatimentioning

confidence: 99%

“…In our case, we will remove the finite subvariety of global minimizers and replace it by second order directional derivatives pointing out from it. These ideas are not new [BSS,Theorem 7.11] but here we add another important idea: We extend the technique of pure states (still real valued!) to work over real closed extension fields of R (see Subsection 2.6 below) rather than over R itself.…”

Section: Systems Of Polynomial Inequalities and Polynomial Optimizatimentioning

confidence: 99%

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On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

Kriel

Schweighofer

2018

Found Comput Math

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Consider a finite system of non-strict polynomial inequalities with solution set S ⊆ R n . Its Lasserre relaxation of degree d is a certain natural linear matrix inequality in the original variables and one additional variable for each nonlinear monomial of degree at most d. It defines a spectrahedron that projects down to a convex semialgebraic set containing S. In the best case, the projection equals the convex hull of S. We show that this is very often the case for sufficiently high d if S is compact and "bulges outwards" on the boundary of its convex hull. Now let additionally a polynomial objective function f be given, i.e., consider a polynomial optimization problem. Its Lasserre relaxation of degree d is now a semidefinite program. In the best case, the optimal values of the polynomial optimization problem and its relaxation agree. We prove that this often happens if S is compact and d exceeds some bound that depends on the description of S and certain characteristicae of f like the mutual distance of its global minimizers on S.

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“…See [1,Hauptsatz] and [15,Théorème 12] for early variants of Theorem 2.5. See [7,Theorem 6.2] and [19,Theorem 5.4.4] for other proofs of Theorem 2.5 in the case d = 1. See Section 6 for the extension of the proof in [19,Theorem 5.4.4] to the case d > 1.…”

Section: Introductionmentioning

confidence: 99%

Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ₁-seminorm Topologies

Ghasemi¹,

Marshall²,

Wagner³

2014

Can. math. bull.

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Abstract. In a paper from 1976, Berg, Christensen and Ressel prove that the closure of the cone of sums of squares R [X] 2 in the polynomial ring R[X] := R[X 1 , . . . , X n ] in the topology induced by the ℓ 1 -norm is equal to Pos ([−1, 1] n ), the cone consisting of all polynomials which are non-negative on the hypercube [−1, 1] n . The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted ℓ 1 -seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi's representation theorem from 2001. At the same time, we use Jacobi's representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer d ≥ 1, the closure of the cone of sums of 2d-powers R [X] 2d in R[X] in the topology induced by the ℓ 1 -norm is equal to Pos ([−1, 1] n ).

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Pure states, nonnegative polynomials and sums of squares

Cited by 17 publications

References 15 publications

A Positivstellensatz for forms on the positive orthant

A Positivstellensatz for forms on the positive orthant

On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ₁-seminorm Topologies

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Pure states, nonnegative polynomials and sums of squares

Cited by 17 publications

References 15 publications

A Positivstellensatz for forms on the positive orthant

A Positivstellensatz for forms on the positive orthant

On the Exactness of Lasserre Relaxations and Pure States Over Real Closed Fields

Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ1-seminorm Topologies

Contact Info

Product

Resources

About

Closure of the Cone of Sums of 2d-powers in Certain Weighted ℓ₁-seminorm Topologies