## Semidefinite Programming for Combinatorial Optimization

Please always quote using this URN: urn:nbn:de:0297-zib-6022

- This book offers a self-contained introduction to the field of semidefinite programming, its applications in combinatorial optimization, and its computational methods. We equip the reader with the basic results from linear algebra on positive semidefinite matrices and the cone spanned by them. Starting from linear programming, we introduce semidefinite programs and discuss the associated duality theory. We then turn to semidefinite relaxations of combinatorial optimization and illustrate their interrelation. In the second half we deal with computational methods for solving semidefinite programs. First, the interior point approach, its iteration complexity, and implementational issues are discussed. Next, we explain in great detail the spectral bundle method, which is particularly suited for large scale semidefinite programming. One of the most successful techniques in integer linear programming is the cutting plane approach which improves an initial relaxation by adding violated inequalities. We explore possibilities to combine the two solution methods with the cutting plane approach in order to strengthen semidefinite relaxations of combinatorial optimization problems.

Author: | Christoph Helmberg |
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Document Type: | Habilitation |

Tag: | combinatorial optimization; max-cut; positive semidefinite matrices; quadrati; semidefinite Programming; semidefinite cone; semidefinite duality |

MSC-Classification: | 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90-02 Research exposition (monographs, survey articles) |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C22 Semidefinite programming | |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C27 Combinatorial optimization | |

Date of first Publication: | 2000/10/10 |

Series (Serial Number): | ZIB-Report (00-34) |

ZIB-Reportnumber: | 00-34 |