### Inhalt des Dokuments

## News

**Registration to the exam**

Some students had problems registering with the examination office, apparently due to some communication problems with that office. This problem should now be resolved: the course is well registered in the list of *Advanced Mathematics Topics*, as this document attests.

**Deadline extension**

The deadline for the programming exercise is extended to Feb. 13, 12:00 (noon)

**The optional programming exercise is online, cf. next paragraph.**

We had a programming session on December 19. Here is the link to the notebook we used for this session. To use this notebook, first go to File > "save a copy in drive". Then, you should be able to modify it and execute the cells.

The solutions to this notebook can be found here (a password was given in class, you can email me if you need it).

### Programming exercises

The link for the programming exercise is here:

Instruction: Copy the notebook to your drive, so you can edit it. Then, fill the TODOs in the notebook. When you are done, download the notebook as a .ipynb file, and send it by email. **The deadline to hand out your .ipynb files is February 13 12:00 (noon).**

# Convex Optimization and Applications (ADM III)

Surprisingly many real-world optimization problems can be reformulated as convex optimization problems. This convexity plays a central role in the computational tractability of a solution. The goals of this course are:

- to provide the students with the necessary background to recognize optimization problems that can be reformulated as convex ones;
- to study the duality theory of convex optimization from the point of view of conic programming, which includes as particular cases the linear programming (LP), semidefinite programming (SDP), second order cone programming (SOCP), and geometric programming (GP);
- to review a variety of applications of convex optimization from various branches such as engineering, control theory, machine learning, robust optimization, and approximation algorithms for combinatorial problems;
- finally, to understand algorithms for convex programming, in particular interior point methods, and to be able to use modern interfaces to pass optimization problems to solvers that implement these algorithms.

# Examination

The oral examinations will take place on **February 28**. **Online** oral exams will take place on **May 14 and May 15**.

Please register for one exam slot at Daniel Schmidt gen. W.'s office (MA 524) during office hours. You also need to register for the exam at the Prüfungsamt, and you can bring us the yellow registration form at the end of any lecture.

All exams will take place in MA 518 *or online (via Zoom)*. Don't forget to bring your student ID!

For the examinations, you are not expected to know everything *by heart. *However, you have to know the basic definitions, know which results exist, and be able to explain them with your own words (there is no need to know very technical formulas, but you must be able to restitute the main message of a theorem, for example. If you ever need a technical formula, there will be a printed copy of the handout available during the exam. You will not be asked to prove results from the course, but rather to demonstrate your general understanding on convex optimization techniques, solve some new exercises, and show your modelling skills.

Feb 28 | May 14 (Online) | May 15 (Online) | July 1 (MA517) | Later | ||||
---|---|---|---|---|---|---|---|---|

Helene | 9:10 | Jan-Erik | 14:00 | Malin | 13:50 | Linda | ||

Felix | 9:50 | Lisa | 14:50 | Anh Huy | 14:50 | Anh Huy | 14:40 | Alfons |

Fatima | 10:30 | Hannah | 15:40 | Diego | 15:40 | Willy | 15:30 | |

Till | 11:10 | Valentin | 16:30 | Franziska | 16:30 | Leonhard | 16:20 | |

Yujin | 11:50 | 17:20 | 17:20 | 17:10 | ||||

Anton | 13:40 | Sebastian | May22 |

## Schedule

Day | Time | Room | |
---|---|---|---|

Lecture | Monday | 10:15 - 11:45 | MA 649 |

Lecture | Thursday | 14:15 - 15:45 | MA 550 |

An exercise session will be held every second week on Thursday.

## Evaluation

**Exercises** will be given on week in advance. At the beginning of exercise sessions, check the exercises you've prepared. One student will be asked to explain his solution. You need 50% of all exercises *checked* to take the exam.

There will be an **oral examination**. You should not learn everything *by heart*, but rather know which result exists. We expect a global understanding of how the chapters of this course articulate together. You will not be asked to prove results from the course, but you will have to solve some new exercises and show your modelling skills.

## References

There will be a handout, posted online as the corresponding chapters are completed.

The course is mainly based on:

- Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge university press.

Selected chapters are also based on the following references:

- "Lecture on Modern Convex Optimization", A. Ben-Tal & A. Nemirovski, 2001.
- "Semidefinite Optimization", Lecture notes of M. Laurent & F. Vallentin at Utrecht.

Further references are indicated directly in the handout.

## Handout

1-Preliminaries | 1_intro.pdf |
---|---|

2-Convex geometry | 2_cvx_geom.pdf |

3-Convex functions | 3_cvx_fun.pdf |

4-Convex optimization | 4_cvx_optim.pdf |

5-Ellipsoid method | 5_ellipsoid.pdf |

6-Conic Programming | 6_conic_programming.pdf |

7-Duality | 7_duality.pdf |

8 - Applications in Combinatorial Optimization | 8_combi.pdf |

9 - Lasserre Hierarchy & Polynomial Optimization | 9_hierarchies.pdf |

10 - Interior points methods | 10_ipm.pdf |

11 - Application to Data Science | 11_data.pdf |

12 - First Order Methods | 12_first_order.pdf |

## Slides

1-Intro | slides-1.pdf |
---|---|

2-Convex geometry | slides-2.pdf |

3-Convex functions | slides-3.pdf |

4-Convex Optimization | slides-4.pdf |

5-Ellipsoid Methods | slides-5.pdf |

8-Combinatorial Optimization | slides-8.pdf |

9-Lasserre Hierarchy | slides-9.pdf |

10-Interior Point Methods | slides-10.pdf |

11-Application to Data Science | slides-11.pdf |

12-First Order Methods | slides-12.pdf |