M2 course at the "Institut Polytechnique de Paris"
Advanced Continuous Optimization

This page describes the 60 hour course given at "Institut Polytechnique de Paris" in the M2 Optimization, during the academic year 2020-2021, entitled Advanced Continuous Optimization. It contains
- the teacher internet pages,
- a short presentation of its contents,
- the detailed program of part I, part II, and part III.

Teachers

Jean Charles Gilbert (Inria Paris),

Presentation

The first part of the module (30 hours) starts with the presentation of the optimality conditions of an optimization problem described in a rather general manner, so that these can be useful for dealing with a large variety of problems: the constraints are expressed by $c(x)\in G$, where $c:\mathbb{E}\to\mathbb{F}$ is a nonlinear map between two Euclidean spaces $\mathbb{E}$ and $\mathbb{F}$, and $G$ is a closed convex part of $\mathbb{F}$. Next, the course describes and analyzes various advanced algorithms to solve functional inclusions (of the kind $F(x)+N(x)\ni0$, where $F$ is a function and $N$ is a multifunction) and optimization problems (nonsmooth methods, linearization methods, proximal and augmented Lagrangian methods, interior point methods) and shows how they can be used to solve a few classical optimization problems (linear optimization, convex quadratic optimization, semidefinite optimization (SDO), nonlinear optimization). Along the way, various tools from convex and nonsmooth analysis will be presented. Everything is conceptualized in finite dimension. The goal of the lectures is therefore to consolidate basic knowledge in optimization, on both theoretical and algorithmic aspects.

The second part of the module (20 hours) focuses on the implementation of some of the previously seen algorithms in Matlab and allows the designer to understand its behavior and to evaluate its efficiency on a concrete problem. A choice will have to be made between the following three projects:

The third part of the module is a 10 hour lecture given by ...

From an academic point of view, the second and third parts are considered to form a same teaching unit, so that they are evaluated together and give rise to a single note. Furthermore, to follow the second and third parts, one must have followed the first part, because the algorithms implemented in the second part are presented and anaylzed in the first part.

Detailed program

First part

Post-lecture notes, slides, and exercises (the access to these notes requires the username "Student" and the given password):

Bibliography

Actual program on a daily basis

Date
Themes of the lectures
Actual details of the contents of the lectures
Monday
November 16
2020
Presentation and recalls Presentation of the course
Background
  • Convex analysis: relative interior, absorbing point, dual cone and Farkas lemma, tangent and normal cones
  • Nonlinear analysis: multifunction
  • Optimization: tangent cone and Peano-Kantorovich necessary optimality condition of the first order for $(P_X)$, sufficient optimality condition of the first order for a convex problem $(P_X)$, KKT conditions for $(P_{EI})$, sufficient conditions for contraint qualification
Monday
November 23
2020
Monday
November 30
2020
Monday
December 7
2020
Monday
December 14
2020
Monday
January 4
2020
Monday
January 11
2020
Monday
January 18
2021
Examination

Second part

It is composed of 5 sessions of 4h each (on Monday 14h-18h), which makes it 20h long.
Examination: ...

The goal of this course is to guide the student in the implementation of some well known optimization algorithms and in its use to solve a concrete problem. The student will choose among the following two projects.

Ensta students having already realized the SQP+HC project must choose another one.

Presentation of the projects ...

Actual program on a daily basis

...

Pieces of code useful for the SQP+HC project: ...

Pieces of code useful for the SDCO+OPF project: ...

Third part

A 10 hour lecture by ...
A short report on these lectures (a PDF file) should be sent before ... at the address Jean-Charles.Gilbert@inria.fr.