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Handbook on Semidefinite, Conic and Polynomial Optimization pp 25–60Cite as

The Approach of Moments for Polynomial Equations

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

Abstract

In this chapter we present the moment based approach for computing all real solutions of a given system of polynomial equations. This approach builds upon a lifting method for constructing semidefinite relaxations of several nonconvex optimization problems, using sums of squares of polynomials and the dual theory of moments. A crucial ingredient is a semidefinite characterization of the real radical ideal, consisting of all polynomials with the same real zero set as the system of polynomials to be solved. Combining this characterization with ideas from commutative algebra, (numerical) linear algebra and semidefinite optimization yields a new class of real algebraic algorithms. This chapter sheds some light on the underlying theory and the link to polynomial optimization.

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Notes

  1. 1.

    A celebrated result of Hilbert (cf. e.g. [2]) shows that there are three sets of parameters (n, d) for which the following equivalence holds: For any polynomial p in n variables and degree 2d, p is nonnegative on \({\mathbb{R}}^{n}\) if and only if p can be written as a sum of squares of polynomials. These parameters are (n = 1, d) (univariate polynomials), (n, d = 1) (quadratic polynomials), and \((n = 3,d = 2)\) (ternary quartic polynomials). In all other cases there are polynomials that are nonnegative on \({\mathbb{R}}^{n}\) but cannot be written as a sum of squares of polynomials.

  2. 2.

    This follows under certain technical conditions on the semidefinite program, which are satisfied for (2.16); see [17] for details.

  3. 3.

    When Λ is positive, the maximality condition on the rank of M⌊t ∕ 2⌋(Λ) implies that the rank of Ms(Λ) is maximum for all s ≤ ⌊t ∕ 2⌋. This is not true for Λ non-positive.

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Authors and Affiliations

  1. Centrum voor Wiskunde en Informatica, Science Park 123, 1098, XG, Amsterdam, The Netherlands

    Monique Laurent

  2. Tilburg University, 90153, 5000, LE, Tilburg, The Netherlands

    Monique Laurent

  3. Institut für Mathematik, Johann Wolfgang Goethe-Universität, Robert-Mayer-Str. 6-10, 60325, Frankfurt am Main, Germany

    Philipp Rostalski

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  1. Monique Laurent
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Correspondence to Monique Laurent .

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Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Laurent, M., Rostalski, P. (2012). The Approach of Moments for Polynomial Equations. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_2

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