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Jean Lasserre
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## Principal Research Fellow

LAAS-CNRS, Toulouse, France

# A new look at nonnegativity on closed sets and polynomial optimization

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### Description

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**Title**: A new look at nonnegativity on closed sets and polynomial optimization

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

**Abstract**: We first show that a continuous function f is nonnegative on a closed set Ksubseteq R^n if and only if (countably many) moment matrices of some signed measure dnu =fdmu with support equal to K, are all positive semidefinite (if K is compact mu is an arbitrary finite Borel measure with support equal to K. In particular, we obtain a convergent explicit hierarchy of semidefinite (outer) approximations with it no lifting, of the cone of nonnegative polynomials of degree at most d. Wen used in polynomial optimization on certain simple closed sets K (like e.g., the whole space R^n, the positive orthant, a box, a simplex, or the vertices of the hypercube), it provides a nonincreasing sequence of upper bounds which converges to the global minimum by solving a hierarchy of semidefinite programs with only one variable. This convergent sequence of upper bounds complements the convergent sequence of lower bounds obtained by solving a hierarchy of semidefinite relaxations.

**Keywords**: optimization control

**Subject**: unspecified

**Area**: Other

**Language**: English

**Year**: 2010

Affiliations : |
LAAS-CNRS, Toulouse, France |

**Journal**: ArXivorg

**Pages**: 1-14

**Url**: http://arxiv.org/abs/1009.0125

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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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