Coursebooks

Optimization on manifolds

MATH-512

Lecturer(s) :

Boumal Nicolas Alain

Language:

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
 - More advanced geometric tools
 - Geodesic convexity

Learning Prerequisites

Required courses


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:

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.

 


Assessment methods

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

Ressources en bibliothèque

In the programs

Reference week

 MoTuWeThFr
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     
Under construction
 
      Lecture
      Exercise, TP
      Project, other

legend

  • Autumn semester
  • Winter sessions
  • Spring semester
  • Summer sessions
  • Lecture in French
  • Lecture in English
  • Lecture in German