Course teacher(s)
Ignace LORIS (Coordinator)ECTS credits
5
Language(s) of instruction
french
Course content
2) Introduction to convex geometry (convex sets, convex hull algorithms, relative interior, projection on a convex set, separating hyperplanes, convex cones)
3) Linear programming (definitions, standard form, two fundamental theorems, basic solutions, simplex algorithm)
4) Duality in linear programming (dual linear programs, weak and strong duality, complimentary slackness, sensitivity analysis, optimal transport)
5) Integer linear programming (definition and example s, maximum weight coupling, totally unimodular matrices, branch and bound algorithm, maximum flow problem, minimal cut problem)
6) Introduction to convex analysis (convex functions, 1st and 2nd order characterisations, continuity and lower semi-continuity, subdifferential, subdifferential calculus)
7) Non linear programming (convex programs, Karush-Kuhn-Tucker condition, Lagrangian duality, sensitivity analysis, application: optimal control)
8) Fenchel transformation (motivation, definition, properties, Fenchel duality, Fenchel-Rockafellar duality, lagrangian duality revisited)
Objectives (and/or specific learning outcomes)
1) model a concrete problem as an optimisation problem
2) understand and use properties of convex sets
3) model problems in terms of linear programs
4) solve simple linear programs with the simplex algorithm
5) understand duality in linear programming
6) formulate and analyze integer linear programmes
7) understand and use the properties of convex functions
8) formulate KKT condtitions for nonlinear problems
9) write the lagrangian and the lagrangian dual problem of a nonlinear problem
10) understand and compute Fenchel transforms
11) identify optimization problems in other disciplines
In the context of the course subjects, a student will be able to interpret and produce mathematical content (such as texts, diagrams, and formulas) using a variety of notations (lowercase and uppercase letters, italics, boldface, Latin and Greek letters, numbers, symbols, etc.), presented in common fonts and sizes.
Prerequisites and Corequisites
Required and Corequired knowledge and skills
analysis (continuity, differentiability, functions of several variables, gradient, open/closed/compact sets...) and linear algebra (vector spaces, matrices, scalar products, ...)
Required and corequired courses
Teaching methods and learning activities
Ex-cathedra course and exercises
References, bibliography, and recommended reading
R. Tyrrell Rockafellar. Convex Analysis. Princeton University Press, 1970.
J. B. Hiriart-Urruty and C. Lemarechal. Convex analysis and minimization algorithms. Springer, 1993.
Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004.
Dimitri P. Bertsekas. Convex Optimization Theory. Athena Scientific, 2009.
Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, 2 edition, 2006.
Amir Beck. Introduction to nonlinear optimization. SIAM, 2014.
Course notes
- Syllabus
- Université virtuelle
Contribution to the teaching profile
1. Acquire and use knowledge
1.1. Learn fundamental concepts in mathematics.
1.2. Assimilate basic notions in algebra, analysis and geometry.
1.3. Analyze, synthesize and link knowledge and different branches of mathematics.
1.4. Master the principles of logical reasoning and use them as the basis of an irrefutable argument .
1.6. Identify an underlying mathematical framework to a given problem.
1.7. Get acquainted with multiple modelisation methods.
2. Understanding and practice of the specifics of a scientific undertaking
2.1. Understand criteria for mathematical rigor, a mathematical argument, methods of proofs.
2.4. Understand the process of studying and modelling data.
2.5. Understand the process of the generalization of a theory.
2.6. Understand the importance of the unification of existing theories.
2.7. Identify questions that occur inside a theory.
2.8. Explore the consequences of a mathematical result.
3. Communication
3.3. Use a clear and rigorous language, adapted to the audience.
4. Ethics and relation to society
4.3. Learn self-criticism with respect to the validity of an argument.
Other information
Additional information
Course notes (pdf) are available (French only) on UV and at PUB.
Contacts
mail (Ignace.Loris@ulb.be), Teams or in teacher's office (campus Plaine, building NO, room 2.O7.107)
Campus
Plaine
Evaluation
Method(s) of evaluation
- written examination
written examination
- Open question with short answer
- Open question with developed answer
- Visual question
- Closed question True or False (T/F)
- Closed question with multiple choices (MCQ)
A single integrated written exam combining theory and exercise questions. Exceptionally (pandemic, open session, ...) the written exan cal be replaced by an oral exam.
Mark calculation method (including weighting of intermediary marks)
A single mark (.../20) will be given at the end of the written exam. There are no partial marks.
Language(s) of evaluation
- french
- (if applicable english )