Study Programmes 2017-2018
WARNING : 2016-2017 version of the course specifications
MATH0462-1  
Discrete optimization
Duration :
30h Th, 20h Pr, 25h Proj.
Number of credits :
Master in biomedical engineering (120 ECTS)5
Master in data science (120 ECTS)5
Master in electrical engineering (120 ECTS)5
Master of science in computer science and engineering (120 ECTS)5
Master in data science and engineering (120 ECTS)5
Master in computer science (120 ECTS)5
Bachelor in mathematics6
Master in mathematics (120 ECTS)6
Lecturer :
Quentin Louveaux
Language(s) of instruction :
English language
Organisation and examination :
Teaching in the first semester, review in January
Units courses prerequisite and corequisite :
Prerequisite or corequisite units are presented within each program
Learning unit contents :
Consider a salesman who must visit 20 potential customers in 20 different cities. A natural question he may ask is to know what is the optimal order in which he has to visit all cities so as to minimze the total distance. This famous problem is better known as the traveling salesman problem. It is the typical example of a discrete optimization problem. Indeed, there is a finite number of solutions (the 20! possible permutations of cities) and we may think of testing them all in order to find the optimal one. This approach is however impossible to perform in practice. Even if we were able to test a billion of these solutions per second, it would take us 77 years to test them all.
The traveling salesman problem is one of many discrete optimization problems. Indeed in particular the problems where binary decisions (such as yes or no) have to be taken often arise in practical applications.
Concerning the contents of the course, as a first part, we concentrate on modeling discrete problems as linear integer programs. We discuss some good principles in order to come up with a formulation. We also see what is needed in order to have a good formulation.
Then the last part of the course deals with the solving techniques of integer programs: mainly branch-and-bound, branch-and-cut, lagrangian relaxation, dynamic programming and approximation algorithms. We also consider some classes of important discrete problems that are well solved, namely flow and matching problems. Finally contraint programming will be briefly considered.
Learning outcomes of the learning unit :
At the end of the course, the student
  • will be able to formulate a real problem as an integer programming model
  • will be able to compare two formulations of a problem
  • will know the main methods to solve integer progamming problems.
Prerequisite knowledge and skills :
A basic course in linear programming.
Planned learning activities and teaching methods :
Traditional tutorials are organized. An implementation project must be achieved.
Mode of delivery (face-to-face ; distance-learning) :
Lecture slides and general information: http://www.montefiore.ulg.ac.be/%7Etcuvelier/math0462
Recommended or required readings :
Two main references are used: For the first part (and the approximation algorithms): D. Bertsimas, R. Weismantel, Optimization over Integers. Dynamic Ideas, 2005. For the second part: L. Wolsey, Integer Programming. Wiley, 1998.
Assessment methods and criteria :
The project grade is the arithmetic mean of the grades of the two projects.
The final exam is written and composed of exercises.
The final grade is the geometric mean of the project grade and of the exam grade.
Work placement(s) :
Organizational remarks :
The course is given in the first semester.
Contacts :