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Sums of Squares, Moment Matrices and Optimization Over Polynomials

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Emerging Applications of Algebraic Geometry

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 149))

Abstract

We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

AMS(MOS) subject classifications. 13P10, 13J25, 13J30, 14PI0, 15A99, 44A60, 90C22, 90C30.

Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203 and by ADONET, Marie Curie Research Training Network MRTN-CT-2003-504438.

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  1. CWI, Kruislaan 413, 1098 SJ, Amsterdam, The Netherlands

    Monique Laurent

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  1. Monique Laurent
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  1. College of Letters & Science, Dept. Mathematics, University of California, Santa Barbara, Santa Barbara, 93106, California, USA

    Mihai Putinar

  2. Department of Mathematics Harvard University, Cambridge, 02138, U.S.A.

    Seth Sullivant

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Laurent, M. (2009). Sums of Squares, Moment Matrices and Optimization Over Polynomials. In: Putinar, M., Sullivant, S. (eds) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09686-5_7

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