### Abstract

We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that the expectation ∫ K f(x)h(x)dx is minimized. We show that the rate of convergence is O(1/r √ ) , where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and K is a full-dimensional compact set satisfying some boundary condition (which is satisfied, e.g., for polytopes and the Euclidean ball). The r-th upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d, and if all moments of order up to 2r+d of the Lebesgue measure on K are known, which holds for example if K is a simplex, hypercube, or a Euclidean ball.

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

Place of Publication | Ithaca |

Publisher | Cornell University Library |

Number of pages | 23 |

Volume | 1411.6867 |

Publication status | Published - 1 Dec 2014 |

### Publication series

Name | Preprint at arXiv |
---|---|

Volume | 1411.6867v3 |

### Keywords

- polynomial optimization
- semidefinite optimization
- Lasserre hierarchy

## Fingerprint Dive into the research topics of 'Convergence Analysis for Lasserre's Measure-based Hierarchy of Upper Bounds for Polynomial Optimization'. Together they form a unique fingerprint.

## Cite this

de Klerk, E., Laurent, M., & Sun, Z. (2014).

*Convergence Analysis for Lasserre's Measure-based Hierarchy of Upper Bounds for Polynomial Optimization*. (Preprint at arXiv; Vol. 1411.6867v3). Cornell University Library. http://arxiv.org/abs/1411.6867