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Article Dans Une Revue SIAM Journal on Optimization Année : 2011

A new look at nonnegativity on closed sets and polynomial optimization

Résumé

We first show that a continuous function f is nonnegative on a closed set $K\subseteq R^n$ if and only if (countably many) moment matrices of some signed measure $d\nu =fd\mu$ 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.
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Dates et versions

hal-00512560 , version 1 (30-08-2010)
hal-00512560 , version 2 (26-04-2011)
hal-00512560 , version 3 (11-05-2011)

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Jean-Bernard Lasserre. A new look at nonnegativity on closed sets and polynomial optimization. SIAM Journal on Optimization, 2011, 21 (3), pp.864-885. ⟨hal-00512560v3⟩
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