### Abstract

Original language | English |
---|---|

Title of host publication | Emerging Applications of Algebraic Geometry |

Editors | M. Putinar, S. Sullivant |

Place of Publication | New York |

Publisher | Springer Verlag |

Pages | 155-270 |

ISBN (Print) | 9780387096858 |

Publication status | Published - 2009 |

### Publication series

Name | The IMA Volumes in Mathematics and its Applications Series |
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Number | 149 |

### Fingerprint

### Cite this

*Emerging Applications of Algebraic Geometry*(pp. 155-270). (The IMA Volumes in Mathematics and its Applications Series; No. 149). New York: Springer Verlag.

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*Emerging Applications of Algebraic Geometry.*The IMA Volumes in Mathematics and its Applications Series, no. 149, Springer Verlag, New York, pp. 155-270.

**Sums of squares, moment matrices and optimization over polynomials.** / Laurent, M.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Scientific › peer-review

TY - CHAP

T1 - Sums of squares, moment matrices and optimization over polynomials

AU - Laurent, M.

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

M3 - Chapter

SN - 9780387096858

T3 - The IMA Volumes in Mathematics and its Applications Series

SP - 155

EP - 270

BT - Emerging Applications of Algebraic Geometry

A2 - Putinar, M.

A2 - Sullivant, S.

PB - Springer Verlag

CY - New York

ER -