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## Principal Research Fellow

LAAS-CNRS, Toulouse, France

# S.o.s. approximation of polynomials nonnegative on a real algebraic set

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**Title**: S.o.s. approximation of polynomials nonnegative on a real algebraic set

**Author(s)**: Jean B Lasserre

**Abstract**: With every real polynomial f, we associate a family fepsilon repsilon, r of real polynomials, in explicit form in terms of f and the parameters epsilon>0,rin N, and such that Vert f-fepsilon rVert1to 0 as epsilonto 0. Let Vsubset R^n be a real algebraic set described by finitely many polynomials equations gj(x)=0,jin J, and let f be a real polynomial, nonnegative on V. We show that for every epsilon>0, there exist nonnegative scalars J such that, for all r sufficiently large, fepsilon r+sumjin J lambdaj(epsilon) gj^2,quad is a sum of squares. This representation is an obvious certificate of nonnegativity of fepsilon r on V, and very specific in terms of the gj that define the set V. In particular, it is valid with it no assumption on V. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of f on a real algebraic set V, or a semi-algebraic set K, and again, with it no assumption on V or K.

**Subject**: unspecified

**Area**: Other

**Language**: English

**Year**: 2004

Affiliations : |
LAAS-CNRS, Toulouse, France |

**Journal**: SIAM Journal on Optimization

**Volume**: 16

**Issue**: 2

**Pages**: 610

**Url**: http://arxiv.org/abs/math/0412400

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- FJean Lasserre,
*Principal Research Fellow*, LAAS-CNRS, Toulouse, France.

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- Jean Lasserre,
*Principal Research Fellow*, LAAS-CNRS, Toulouse, France.

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