### Abstract

models broad classes of hard problems from combinatorics, geometry and optimization.

New algorithmic approaches have emerged recently for computing the global

minimum, by combining tools from real algebra (sums of squares of polynomials) and functional

analysis (moments of measures) with semidefinite optimization. Sums of squares

are used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies to

some combinatorial graph problems, and discuss links with other relaxation methods.

Original language | English |
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Title of host publication | Proceedings of the International Congress of Mathematicians 2014 (ICM 2014) |

Editors | Sun Young Jang, Young Rock Kim, Dae-Woong Lee, Ikkwon Yie |

Place of Publication | Seoul |

Publisher | Kyung Moon SA Co. Ltd. |

Pages | 843-869 |

ISBN (Print) | 9788961058070 |

Publication status | Published - 2014 |

Event | International Congress of Mathematicians 2014 - Seoul, Korea, Republic of Duration: 13 Aug 2014 → 21 Aug 2014 |

### Conference

Conference | International Congress of Mathematicians 2014 |
---|---|

Abbreviated title | ICM 2014 |

Country | Korea, Republic of |

City | Seoul |

Period | 13/08/14 → 21/08/14 |

### Fingerprint

### Cite this

*Proceedings of the International Congress of Mathematicians 2014 (ICM 2014)*(pp. 843-869). Seoul: Kyung Moon SA Co. Ltd..

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*Proceedings of the International Congress of Mathematicians 2014 (ICM 2014).*Kyung Moon SA Co. Ltd., Seoul, pp. 843-869, International Congress of Mathematicians 2014, Seoul, Korea, Republic of, 13/08/14.

**Optimization over polynomials : Selected topics.** / Laurent, M.

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

TY - GEN

T1 - Optimization over polynomials

T2 - Selected topics

AU - Laurent, M.

PY - 2014

Y1 - 2014

N2 - Minimizing a polynomial function over a region defined by polynomial inequalitiesmodels broad classes of hard problems from combinatorics, geometry and optimization.New algorithmic approaches have emerged recently for computing the globalminimum, by combining tools from real algebra (sums of squares of polynomials) and functionalanalysis (moments of measures) with semidefinite optimization. Sums of squaresare used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies tosome combinatorial graph problems, and discuss links with other relaxation methods.

AB - Minimizing a polynomial function over a region defined by polynomial inequalitiesmodels broad classes of hard problems from combinatorics, geometry and optimization.New algorithmic approaches have emerged recently for computing the globalminimum, by combining tools from real algebra (sums of squares of polynomials) and functionalanalysis (moments of measures) with semidefinite optimization. Sums of squaresare used to certify positive polynomials, combining an old idea of Hilbert with the recent algorithmic insight that they can be checked efficiently with semidefinite optimization. The dual approach revisits the classical moment problem and leads to algorithmic methods for checking optimality of semidefinite relaxations and extracting global minimizers. We review some selected features of this general methodology, illustrate how it applies tosome combinatorial graph problems, and discuss links with other relaxation methods.

M3 - Conference contribution

SN - 9788961058070

SP - 843

EP - 869

BT - Proceedings of the International Congress of Mathematicians 2014 (ICM 2014)

A2 - Jang, Sun Young

A2 - Kim, Young Rock

A2 - Lee, Dae-Woong

A2 - Yie, Ikkwon

PB - Kyung Moon SA Co. Ltd.

CY - Seoul

ER -