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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polynomials nonnegative on a grid and discrete optimization


Author: Jean B. Lasserre
Journal: Trans. Amer. Math. Soc. 354 (2002), 631-649
MSC (2000): Primary 13P99, 30C10; Secondary 90C10, 90C22
Published electronically: October 4, 2001
MathSciNet review: 1862561
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Abstract: We characterize the real-valued polynomials on $\mathbb R^n$that are nonnegative (not necessarily strictly positive) on a grid $\mathbb K$ of points of $\mathbb R^n$, in terms of a weighted sum of squares whose degree is bounded and known in advance. We also show that the mimimization of an arbitrary polynomial on $\mathbb K$ (a discrete optimization problem) reduces to a convex continuous optimization problem of fixed size. The case of concave polynomials is also investigated. The proof is based on a recent result of Curto and Fialkow on the $\mathbb K$-moment problem.


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Additional Information

Jean B. Lasserre
Affiliation: LAAS-CNRS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex, France
Email: lasserre@laas.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02898-7
PII: S 0002-9947(01)02898-7
Keywords: Algebraic geometry; positive polynomials; $\K$-moment problem; semidefinite programming
Received by editor(s): November 11, 2000
Published electronically: October 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society