## Abstract

In this paper, we consider approximation algorithms for optimizing a generic multivariate polynomial function in discrete (typically binary) variables. Such models have natural applications in graph theory, neural networks, error-correcting codes, among many others. In particular, we focus on three types of optimization models: (1) maximizing a homogeneous polynomial function in binary variables; (2) maximizing a homogeneous polynomial function in binary variables, mixed with variables under spherical constraints; (3) maximizing an inhomogeneous polynomial function in binary variables. We propose polynomial-time randomized approximation algorithms for such polynomial optimization models, and establish the approximation ratios (or relative approximation ratios whenever appropriate) for the proposed algorithms. Some examples of applications for these models and algorithms are discussed as well.

Original language | English (US) |
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Pages (from-to) | 3-36 |

Number of pages | 34 |

Journal | Journal of the Operations Research Society of China |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2013 |

### Bibliographical note

Funding Information:Simai He was supported in part by Hong Kong General Research Fund (No. CityU143711). Zhening Li was supported in part by Natural Science Foundation of Shanghai (No. 12ZR1410100) and Ph.D. Programs Foundation of Chinese Ministry of Education (No. 20123108120002). Shuzhong Zhang was supported in part by U.S. National Science Foundation (No. CMMI-1161242).

## Keywords

- Approximation algorithm
- Approximation ratio
- Binary integer programming
- Mixed integer programming
- Polynomial optimization problem