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Cours scientifiques - MAP554D : Optimisation (M1)

Domaine > Mathématiques appliquées.

Descriptif

Catalog Description

This course provides a comprehensive introduction to the theory and applications of Optimisation. It builds upon fundamental concepts in linear algebra, calculus, and numerical analysis. The students will learn the basic concepts, tools, and methods used in Optimisation as well as their employment on some real-life applications.

Goal

This course aims to familiarize students with the main concepts, tools, and methods of mathematical optimisation. This understanding will enable the students with some of the necessary mathematical background to handle other courses in this master’s program and to approach different real-life applications that can be written as optimisation problems.

Format des notes

Numérique sur 20

Littérale/grade réduit

Pour les étudiants du diplôme M1 Mathématiques Appliquées et Statistiques

Programme détaillé

Assessment

Final exam (75%) and mid-term project (25%).

 

Teaching and Learning Methodologies

The course will be taught through in-class lectures, problem-solving sessions and coursework.

 

Course Learning Outcomes (CLOs)

  1. Master a range of concepts, theories and methods for dealing with various classes of optimization problems.
  2. Demonstrate a deep understanding of the mathematical methods for approaching optimization problems.
  3. Express advanced problem-solving skills by independently applying mathematical principles to solve real life applications modeled as optimization problems.
  4. Develop advanced abilities in abstract thinking, spatial imagination, logical reasoning and judgment.

 

Schedule of Laboratory and Other Non-Lecture Sessions

There will be one tutorial session each week, in which the professor will provide students with the necessary support for enabling them to complete the coursework. Depending on the availability of resources, some tutorial sessions might be held at the computer in order to facilitate the students to implement the discussed algorithms.

 

Teaching Plan Overview

The course is taught with a weekly lecture of 2 hours. The lecturer will introduce a topic in detail, engage students in interactive discussions, and show demonstrations (where relevant) to reinforce the lecture content. The lecturer will also provide (where relevant) handouts, and references to reading material, where necessary. Each lecture will be accompanied by a 2-hour tutorial session, where the theoretical achievements will be illustrated with exercises and applications.

 

Teaching Plan

Week 1. Introduction: classes of optimization problems, examples, solution notions

Week 2. Linear structures: linear equality systems, methods for solving them

Week 3. Convexity: convex sets and functions, properties, characterizations

Week 4. Alternatives: Farkas’ Lemma, theorems of the alternative

Week 5. Unconstrained differentiable optimization problems: necessary and sufficient optimality conditions, Fermat rule

Week 6. Constrained differentiable optimization problems: necessary and sufficient optimality conditions, Karush-Kuhn-Tucker system

Week 7. Unconstrained differentiable optimization problems: descent algorithms

Week 8. Unconstrained differentiable optimization problems: Newton’s algorithm, quasi-Newton methods

Week 9. Constrained differentiable optimization problems: algorithms

Week 10. Constrained differentiable optimization problems: Sequential Quadratic Programming

Week 11. Conjugacy and subdifferentiability: properties, characterizations, Fenchel-Moreau statement

Week 12. Convex optimization problems: properties, approaches

Week 13. Convex optimization problems: duality, optimality conditions

Week 14. Convex optimization problems: algorithms, subgradient methods

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