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On the complexity of Putinar's Positivstellensatz | | Jiawang Nie
; Markus Schweighofer
; | | Date: |
14 Oct 2005 | | Abstract: | Let $S={xinR^nmid g_1(x)ge 0,...,g_m(x)ge 0}$ be a basic closed semialgebraic set defined by real polynomials $g_i$. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of $S$, every real polynomial $f$ positive on $S$ posesses a representation $f=sum_{i=0}^msi_ig_i$ where $g_0:=1$ and each $si_i$ is a sum of squares of polynomials. Such a representation is a certificate for the nonnegativity of $f$ on $S$. We give a bound on the degrees of the terms $si_ig_i$ in this representation which depends on the description of $S$, the degree of $f$ and a measure of how close $f$ is to having a zero on $S$. As a consequence, we get information about the convergence rate of Lasserre’s procedure for optimization of a polynomial subject to polynomial constraints. | | Source: | arXiv, math.AG/0510309 | | Other source: | [GID 502219] math/0510309 | | Services: | Forum | Review | PDF | Favorites | |
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