### Abstract

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

Pages (from-to) | 433-457 |

Number of pages | 24 |

Journal | Mathematical Programming |

Volume | 151 |

Issue number | 2 |

Early online date | 15 Oct 2014 |

DOIs | |

Publication status | Published - Jul 2015 |

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### Keywords

- polynomial optimization over the simplex
- PTAS
- Bernstein approximation

### Cite this

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*Mathematical Programming*, vol. 151, no. 2, pp. 433-457. https://doi.org/10.1007/s10107-014-0825-6

**An alternative proof of a PTAS for fixed-degree polynomial optimization over the simplex.** / de Klerk, E.; Laurent, M.; Sun, Z.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - An alternative proof of a PTAS for fixed-degree polynomial optimization over the simplex

AU - de Klerk, E.

AU - Laurent, M.

AU - Sun, Z.

PY - 2015/7

Y1 - 2015/7

N2 - The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems—it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a polynomial-time approximation scheme (PTAS) for polynomials of fixed degree, which is based on polynomial evaluations at the points of a sequence of regular grids. In this paper, we provide an alternative proof of the PTAS property. The proof relies on the properties of Bernstein approximation on the simplex. We also refine a known error bound for the scheme for polynomials of degree three. The main contribution of the paper is to provide new insight into the PTAS by establishing precise links with Bernstein approximation and the multinomial distribution.

AB - The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems—it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a polynomial-time approximation scheme (PTAS) for polynomials of fixed degree, which is based on polynomial evaluations at the points of a sequence of regular grids. In this paper, we provide an alternative proof of the PTAS property. The proof relies on the properties of Bernstein approximation on the simplex. We also refine a known error bound for the scheme for polynomials of degree three. The main contribution of the paper is to provide new insight into the PTAS by establishing precise links with Bernstein approximation and the multinomial distribution.

KW - polynomial optimization over the simplex

KW - PTAS

KW - Bernstein approximation

U2 - 10.1007/s10107-014-0825-6

DO - 10.1007/s10107-014-0825-6

M3 - Article

VL - 151

SP - 433

EP - 457

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 2

ER -