Lower bounds for a polynomial in terms of its coefficients

Ghasemi, Mehdi; Marshall, Murray

doi:10.1007/s00013-010-0179-0

Cited by 9 publications

(11 citation statements)

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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%

“…Examples(3)and(4)can be seen as special cases of (6): If each I j is singleton, (6) produces the lower bound for f onn i=1 [−N i , N i ] described in(4). If there is just one I j , i.e., m = 1 and I 1 = {1, .…”

mentioning

confidence: 99%

“…. , I m } and polynomials f , each f having t terms and deg(f ) ≤ d with coefficients chosen from [−10, 10] 4. See also[3,…”

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confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

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Lower Bounds for Polynomials Using Geometric Programming

Ghasemi¹,

Marshall²

2012

SIAM J. Optim.

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“…Positivity certificates and relaxations for polynomial optimization problems. Problem (1.1) falls in the general class of polynomial optimization problems, where the hypercube Q is replaced by an arbitrary compact basic closed semialgebraic set 4) and g j ∈ R[x] are given polynomials. We find the hypercube S = Q when considering the s = 2n (linear) polynomials…”

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confidence: 99%

Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube

Klerk¹,

Laurent²

2010

SIAM J. Optim.

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Abstract. We consider the problem of minimizing a polynomial on the hypercube [0, 1] n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmüdgen [26]. The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the hypercube.

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“…[7]). This limitation of semidefinite programming to implement sums of squares (SOS) representations, was the motivation for providing other SOS certificates and yielded the sufficient conditions of [4] and subsequently of [2,5]. And in a recent work Ghasemi and Marshall [6] have shown how to compute a lower bound on the global optimum of a multivariate polynomial on R n , by solving a certain geometric program.…”

Section: Introductionmentioning

confidence: 99%

Lower bounds on the global minimum of a polynomial

2013

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We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$ on the closed ball ${x \in \mathbb{R}^n : \sum x_i^{2d} \le M}$, computable by geometric programming, for any real $M$. We compare this bound with the (global) lower bound $f_{{\rm gp}}$ obtained by Ghasemi and Marshall, and also with the hierarchy of lower bounds, computable by semidefinite programming, obtained by Lasserre [SIAM J. Opt. 11(3) (2001) pp 796-816]. Our computations show that the bound $f_{{\rm gp},M}$ improves on the bound $f_{{\rm gp}}$ and that the computation of $f_{{\rm gp},M}$, like that of $f_{{\rm gp}}$, can be carried out quickly and easily for polynomials having of large number of variables and/or large degree, assuming a reasonable sparsity of coefficients, cases where the corresponding computation using semidefinite programming breaks down

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Lower bounds for a polynomial in terms of its coefficients

Cited by 9 publications

References 9 publications

Lower Bounds for Polynomials Using Geometric Programming

Lower Bounds for Polynomials Using Geometric Programming

Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube

Lower bounds on the global minimum of a polynomial

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