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Abstract: 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 signedmeasure $d u =fd\mu$ with support equal to K, are all positive semidefiniteif $K$ is compact $\mu$ is an arbitrary finite Borel measure with supportequal to K. In particular, we obtain a convergent explicit hierarchy ofsemidefinite outer approximations with {\it no} lifting, of the cone ofnonnegative polynomials of degree at most $d$. Wen used in polynomialoptimization 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 thehypercube, it provides a nonincreasing sequence of upper bounds whichconverges to the global minimum by solving a hierarchy of semidefinite programswith only one variable. This convergent sequence of upper bounds complementsthe convergent sequence of lower bounds obtained by solving a hierarchy ofsemidefinite relaxations.



Autor: Jean B. Lasserre LAAS

Fuente: https://arxiv.org/







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