### Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Recitations: 1 session / week, 1 hour / session

### Course Content

The course is a comprehensive introduction to the theory, algorithms and applications of integer optimization and is organized in five parts.

#### Part I Formulations, complexity and relaxations, Lectures 1-10

Discusses how to formulate integer optimization problems, how to enhance the formulations to improve the quality of relaxations, how to obtain ideal formulations, the duality of integer optimization and how to solve the resulting relaxations both practically and theoretically. It also gives insight on why integer optimization problems are difficult.

#### Part II Robust discrete optimization, Lecture 11

Treats robust discrete optimization. This is a tractable methodology to address problems under uncertainty.

#### Part III Algebra and geometry of integer optimization, Lectures 12-16

Develops the theory of lattices, outlines ideas from algebraic geometry that have had an impact on integer optimization, and discusses the geometry of integer optimization. These lectures provide the building blocks for developing algorithms.

#### Part IV Algorithms for integer optimization, Lectures 17-23

Develops cutting plane methods, enumerative and heuristic methods and approximation algorithms.

#### Part V Mixed integer optimization, Lectures 24-25

Treats mixed integer optimization. This is a practically significant area as real world problems have very often both continuous and discrete variables.

### Required Textbook

Bertsimas, Dimitris, and Robert Weismantel. *Optimization over Integers*. Belmont, MA: Dynamic Ideas, 2005. ISBN: 9780975914625.

### Grading

Grades will be determined by performance on the following requirements. Weights are approximate, and class participation is an important tie breaker.

ACTIVITIES | PERCENTAGES |
---|---|

Homework | 30% |

Midterm exam | 30% |

Final exam | 40% |

### Calendar

LEC # | TOPICS |
---|---|

1 | Formulations |

2 | Complexity |

3-4 | Methods to enhance formulations |

5-7 | Ideal formulations |

8-9 | Duality theory |

10 | Algorithms for solving relaxations |

11 | Robust discrete optimization |

12-13 | Lattices |

Midterm exam | |

14-15 | Algebraic geometry |

16 | Geometry |

17-18 | Cutting plane methods |

19 | Enumerative methods |

20 | Heuristic methods |

21-23 | Approximation algorithms |

24-25 | Mixed integer optimization |

Final exam |