### Abstract

The generalized problem of moments is a conic linear optimization

problem over the convex cone of positive Borel measures with given support. It

has a large variety of applications, including global optimization of polynomials

and rational functions, option pricing in finance, constructing quadrature schemes

for numerical integration, and distributionally robust optimization. A usual solution

approach, due to J.B. Lasserre, is to approximate the convex cone of positive

Borel measures by finite dimensional outer and inner conic approximations. We

will review some results on these approximations, with a special focus on the

convergence rate of the hierarchies of upper and lower bounds for the general

problem of moments that are obtained from these inner and outer approximations.

problem over the convex cone of positive Borel measures with given support. It

has a large variety of applications, including global optimization of polynomials

and rational functions, option pricing in finance, constructing quadrature schemes

for numerical integration, and distributionally robust optimization. A usual solution

approach, due to J.B. Lasserre, is to approximate the convex cone of positive

Borel measures by finite dimensional outer and inner conic approximations. We

will review some results on these approximations, with a special focus on the

convergence rate of the hierarchies of upper and lower bounds for the general

problem of moments that are obtained from these inner and outer approximations.

Original language | English |
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Title of host publication | World Women in Mathematics 2018 |

Subtitle of host publication | Proceedings of the First World Meeting for Women in Mathematics (WM)² |

Editors | Carolina Araujo, Georgia Benkart, Cheryl E. Praeger, Betül Tanbay |

Place of Publication | Cham |

Publisher | Springer |

Pages | 17-56 |

ISBN (Print) | 9783030211691 |

DOIs | |

Publication status | Published - Dec 2019 |

### Publication series

Name | Association for Women in Mathematics Series |
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Publisher | Springer |

Volume | 20 |

### Fingerprint

### Cite this

de Klerk, E., & Laurent, M. (2019). A survey of semidefinite programming approaches to the generalized problem of moments and their error analysis. In C. Araujo, G. Benkart, C. E. Praeger, & B. Tanbay (Eds.),

*World Women in Mathematics 2018: Proceedings of the First World Meeting for Women in Mathematics (WM)²*(pp. 17-56). (Association for Women in Mathematics Series ; Vol. 20). Springer. https://doi.org/10.1007/978-3-030-21170-7_1