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 | References | Similar Articles | Additional Information

Abstract: We characterize the real-valued polynomials on that are nonnegative (not necessarily strictly positive) on a grid of points of , 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 (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 -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

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