Linear and Discrete Optimization

apply

Next Session:

Feb 18th 2013 (7 weeks long)

Workload: 4-6 hours/week 

About the Course

This course serves as an introduction to linear and discrete optimization from the viewpoint of a mathematician or computer scientist.  Besides learning how linear and discrete optimization can be applied, we focus on understanding methods that solve linear programs and discrete optimization problems in a mathematically rigorous way.

We will answer  questions like:

• Does a particular method work correctly?
• Does it terminate and, if yes, in what time?
• Can we prove that a solution is optimal?

The course starts by discussing what a linear program is and how linear programming can be applied. Then, we will treat the simplex method and the theory of duality. We will also discuss integer programming problems where the variables are constrained to integer values. Finally we will address the question of whether linear programs can be efficiently solved in theory which leads us to the ellipsoid method.

The course constitutes about half of the material on linear and discrete optimization that is taught for mathematics and computer science undergraduates at EPFL and will feature video lectures, quizzes, programming assignments, and a final exam. 

About the Instructor(s)

Friedrich Eisenbrand is a professor of mathematics and (by courtesy) computer science at EPFL. His main research interests lie in the field of discrete optimization, in particular in algorithms and complexity, integer programming, geometry of numbers, and applied optimization. He is best known for his work on efficient algorithms for integer programming in fixed dimension and the theory of cutting planes, which are an important tool to solve large scale industrial optimization problems.

Course Syllabus

1. Linear programming, modeling, equivalence of standard forms
2. Basic solutions, primal and dual feasible basic solutions, pivoting and the simplex method
3. Termination and complexity of the simplex method
4. Integer programming, bipartite matching and flows
5. Models of computation, bit-complexity
6. Convex sets, separation theorem and convex optimization
7. Linear programming in polynomial time: The Ellipsoid method

Recommended Background

The most important prerequisite is familiarity with linear algebra. It will be useful to have some programming proficiency in a higher level programming language such as Java or Python. 

Suggested Readings

Although the course is self-contained and  the lecture notes will be available on-line, it will be useful to consult further literature. There are many excellent texts on discrete optimization. Among our favorites are the books Understanding and Using Linear Programming, by Matousek and Gärtner, Introduction to Linear Optimization by Bertsimas and Tsitsiklis, and Theory of Linear and Integer Programming  by Schrijver.

Course Format

The class consists of lecture videos punctuated by quizzes. There will also be standalone homeworks that are not part of the video lectures, programming assignments, and a final exam.

FAQ

Will I get a certificate for completing this class? 
Students who successfully complete the class (80% of assignments and final exam) will receive a certificate signed by the instructor. 

What resources will I need for the class? 
For this course, all you need is an Internet connection, and the time to read and think.

What are the prerequisites for this class? 
The most important prerequisites are solid knowledge of linear algebra and some programming proficiency in a higher level programming language such as Java or Python.