Master M2 MVA: Convex Optimization, Algorithms and Applications.

Description

The objective of this course is to learn to recognize, transform and solve a broad class of convex optimization problems arising in various fields such as machine learning, finance or signal processing. The course starts with a basic primer on convex analysis followed by a quick overview of convex duality theory. The second half of the course is focused on algorithms, including first-order and interior point methods, together with bounds on their complexity. The course ends with illustrations of these techniques in various applications.

Course organization

• Location and time: 13:00 - 16:00, amphi Alain Aspect (1G58), ENS Paris Saclay.

  • Sep. 2026: 28

  • Oct. 2026: 5, 12

  • Nov. 2026: 2, 9, 16.

Organisation

The course will be live but course videos from 2020 are available below.

• Introduction (40 minutes)

• Convex Sets (32 minutes)

• Convex Functions (49 minutes)

• Convex Optimization Problems (51 minutes)

• Duality (120 minutes)

• Unconstrained Optimization (59 minutes)

• Equality Constrained Optimization (29 minutes)

• Barrier Method (53 minutes)

• First Order Methods Part I (33 minutes)

• First Order Methods Part II (47 minutes)

• Applications (55 minutes)

• SDP Applications (26 minutes)

• l1 Penalties (16 minutes)

Notes

• Intro & convex sets

• Convex functions and problems

• Duality

• Newton's method

• Interior point methods

• First order methods I

• First order methods II

• Applications I

• Applications II

• SDP Applications

• Compressed Sensing

• Stochastic and randomized convex optimization (Taught by Adrien Taylor)

Program

The course is split in three parts.

• Modeling

  • Convex sets, functions and problems

  • Duality

• Algorithms

  • Interior point methods

  • Complexity

  • First-order methods, acceleration

• Applications

  • Machine learning and statistics

  • Signal processing

  • Combinatorial problems

  • Finance

References

• Convex Optimization, S. Boyd and L. Vandenberghe, Cambridge University Press.

• Introductory Lectures on Convex Optimization, Y. Nesterov, Springer.

• Lectures on Modern Convex Optimization, A. Nemirovski and A. Ben-Tal, SIAM.

Exercices

Many of the exercises are taken from the textbook by Boyd et Vandenberghe. We will be using Gradescope to grade assignments.

Exam

Final exam, on Monday Nov. 30 2026, 13:00-16:00, ENS Paris-Saclay, salle 1G58 (grand amphi Alain Aspect).
Closed book. You are allowed one page of notes (recto-verso). The exam will be mostly focused on chapters 1-5 of the textbook (up to duality) Final exam 2016.