Sufficient conditions for a real polynomial to be a sum of squares

Lasserre, Jean-Bernard

doi:10.1007/s00013-007-2251-y

Cited by 15 publications

(14 citation statements)

References 9 publications

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“…Then observe that in Theorem 8, and with r = max Hence, an important issue to find sufficient conditions to ensure that the Lagrangian L f is s.o.s., and if possible, conditions that can be checked directly from the data g j . 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.…”

Section: Sdr For Compact Convex Basic Semialgebraic Setsmentioning

confidence: 99%

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: Sdr For Compact Convex Basic Semialgebraic Setsmentioning

confidence: 99%

Convex sets with semidefinite representation

Lasserre

2008

Math. Program.

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“…Let with the case m = 0, producing a lower bound f gp for f on R n computable by geometric programming. 1 See [2], [4] and [7] for precursors of [5]. The algorithm in [3] is a variation of the algorithm in [5], which deals with the case m = 1, g 1 = M − (x d 1 + · · · + x d n ), i.e., it produces a lower bound for f on the hyperellipsoid…”

Section: Introductionmentioning

confidence: 99%

“…where ∆ (f ) := {α ∈ N n : |α| > 0 and f α x α is not a square in R[x]} and N α := n i=1 N αi i . Set (7) f tr,N :…”

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confidence: 99%

Lower Bounds for Polynomials Using Geometric Programming

Ghasemi¹,

Marshall²

2012

SIAM J. Optim.

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Abstract. We make use of a result of Hurwitz and Reznick [8] [19], and a consequence of this result due to Fidalgo and Kovacec [5], to determine a new sufficient condition for a polynomial f ∈ R[X 1 , . . . , X n ] of even degree to be a sum of squares. This result generalizes a result of Lasserre in [10] and a result of Fidalgo and Kovacec in [5], and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound f gp for f , and we explain how fgp can be computed using geometric programming. The lower bound f gp is generally not as good as the lower bound f sos introduced by Lasserre [11] and Parrilo and Sturmfels [15], which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of f gp is much faster. The computation is simplest when the highest degree term of f has the form. . , n. The lower bounds for f established in [6] are obtained by evaluating the objective function of the geometric program at the appropriate feasible points.

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“…We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[X] of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec [2] and of Lasserre [6]. Exploiting these results, we determine, for any polynomial f ∈ R[X] of degree 2d whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f − r is a sum of squares of polynomials.…”

mentioning

confidence: 79%

“…In this paper we are interested in some recent results, due to Lasserre [6] and to Fidalgo and Kovacec [2], which give sufficient conditions on the coefficients for a polynomial to be a sum of squares. We establish new and improved versions of these results; see Ths.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds for a polynomial in terms of its coefficients

Ghasemi

Marshall

2010

Arch. Math.

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Abstract. We determine new sufficient conditions in terms of the coefficients for a polynomial f ∈ R[X] of degree 2d (d ≥ 1) in n ≥ 1 variables to be a sum of squares of polynomials, thereby strengthening results of Fidalgo and Kovacec [2] and of Lasserre [6]. Exploiting these results, we determine, for any polynomial f ∈ R[X] of degree 2d whose highest degree term is an interior point in the cone of sums of squares of forms of degree d, a real number r such that f − r is a sum of squares of polynomials. The existence of such a number r was proved earlier by Marshall [8], but no estimates for r were given. We also determine a lower bound for any polynomial f whose highest degree term is positive definite.

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Sufficient conditions for a real polynomial to be a sum of squares

Cited by 15 publications

References 9 publications

Convex sets with semidefinite representation

Convex sets with semidefinite representation

Lower Bounds for Polynomials Using Geometric Programming

Lower bounds for a polynomial in terms of its coefficients

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