# Fiches de cours

## Optimization on manifolds

#### Enseignant(s) :

Boumal Nicolas Alain

English

#### Summary

We develop, analyze and implement numerical algorithms to solve optimization problems of the form: min f(x) where x is a point on a smooth manifold. To this end, we first study differential and Riemannian geometry (with a focus dictated by pragmatic concerns). We also discuss several applications.

#### Content

- Applications of optimization on manifolds
- First-order Riemannian geometry in Euclidean spaces
- First-order optimization algorithms on manifolds
- Second-order Riemannian geometry in Euclidean spaces
- Second-order optimization algorithms on manifolds
- Fundamentals of differential geometry (general framework)
- Riemannian quotient manifolds
- Geodesic convexity

#### Learning Prerequisites

##### Required courses

•  Analysis
•  Linear algebra
•  Elements of numerical linear algebra and numerical methods
•  Programming skills in a language suitable for scientific computation (Matlab, Python, Julia...)

There are no prerequisites in differential or Riemannian geometry, or in optimization. Students should be comfortable with proofs.

#### Learning Outcomes

By the end of the course, the student must be able to:
• Manipulate concepts of differential and Riemannian geometry.
• Develop geometric tools to work on new manifolds of interest.
• Recognize and formulate a Riemannian optimization problem.
• Analyze implement and compare several Riemannian optimization algorithms.
• Apply the general theory to particular cases.
• Prove some of the most important theorems studied in class.

#### Teaching methods

Lectures + exercise sessions

#### Expected student activities

Students are expected to attend lectures and participate actively in class and exercises. Exercises will include both theoretical work and programming assignments. Students also complete projects that likewise include theoretical and numerical work.

Projects

#### Resources

##### Bibliography

Lecture notes: "An introduction to optimization on smooth manifolds", available online: http://www.nicolasboumal.net/book
- Book: "Optimization algorithms on matrix manifolds", P.-A. Absil, R. Mahoney and R. Sepulchre, Princeton University Press 2008: https://press.princeton.edu/absil
- Book "Introduction to Smooth Manifolds", John M. Lee, Springer 2012: https://link.springer.com/book/10.1007/978-1-4419-9982-5
- Book "Introduction to Riemannian Manifolds", John M. Lee, Springer 2018: https://link.springer.com/book/10.1007/978-3-319-91755-9

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand