Saarland University, Machine Learning Group, Fak. MI - Mathematik und Informatik, Campus E1 1, 66123 Saarbr├╝cken, Germany     

Machine Learning Group
Department of Mathematics and Computer Science - Saarland University



Sommersemester 2012


Convex optimization problems arise quite naturally in many application areas like signal processing, machine learning, image processing, communication and networks and finance etc.

The course will give an introduction into convex analysis, the theory of convex optimization such as duality theory, algorithms for solving convex optimization problems such as interior point methods but also the basic methods in general nonlinear unconstrained minimization, and recent first-order methods in non-smooth convex optimization. If time allows I will also cover d.c. (difference of convex) programming and convex problems as relaxations of hard combinatorial problems. While the emphasis is given on mathematical and algorithmic foundations, several example applications will be discussed.

The course requires a good background in linear algebra and calculus, but no prior knowledge in optimization is required. The course will follow in the first part mainly the book of Boyd and Vandenberge on "Convex Optimization" and can be seen as complementary to the core lecture "Optimization" which will also takes place during the summer semester.

Students who intend to do their master thesis in machine learning are encouraged to take this course.

Type: Advanced course (Vertiefungsvorlesung), 6 credit points


The course follows in the first part the book of Boyd and Vandenberghe.

The practical exercises will be in Matlab and will make use of CVX.


Slides 1 (Introduction/Reminder LA and Analysis)

Exercise 1 Solution 1
Slides 2 (Convex Sets)

Exercise 2 Solution 2
Slides 3 (Convex Functions)

Exercise 3 Solution 3
Slides 4 (Convex Optimization)

Exercise 4 Solution 4
Slides 5 (Convex Optimization ctd)

Exercise 5 Solution 5
Slides 6 (Optimality Conditions)

Exercise 6 Solution 6
Slides 7 (Optimality Conditions ctd)

Exercise 7 Solution 7
Slides 8 (Unconstrained Minimization)

Exercise 8 Matlab Files Solution 8
Slides 9 (Constrained Minimization)

Exercise 9 Matlab Files Solution 9
Slides 10 (Barrier Method)

Exercise 10 SVM Data Solution10
Slides 11 (Constrained First Order Methods)

Exercise 11 Solution11
Slides 12 (Proximal Gradient Methods)

Exercise 12 TV Data Solution12
Slides 13 (Non-convex Problems)



  • The lecture will be based on the following book:
    S. Boyd and L. Vandenberghe: Convex Optimization, Cambridge University Press, (2004).
    The book is freely available
  • Complementary book: D. P. Bertsekas: Nonlinear Programming, Athena Scientific, (1999).
  • Other resources:
    • Matlab is available on cip[101-114] and cip[220-238], gpool[01-27]
      The path is /usr/local/matlab/bin.
      For the sun workstations you have to select in the menu Applications/studcsApplications/Matlab
      Access from outside should be possible via ssh: ssh -X
    • Matlab tutorial by David F. Griffiths  


Oral Exam Schedule: if the scheduled time is not possible for you, write an email to Prof. Hein

  • Thursday, 10.00, Pavel Kolev,
  • Thursday, 11.00, Anastasia Podosinnikova,
  • Thursday, 14.00, Bernhard Schommer,
  • Thursday, 16.00, Pramod Kaushik Mudrakarta,
  • Friday, 10.00, Pedro Mercado,
  • Friday, 11.00, Joanner Pacia,
  • Friday, 14.00, Sanjar Karaev,
  • Friday, 15.00, Mohamed Omran,
  • Friday, 16.00, Rui Xu,


Lecture: Tuesday, 14-16, E1 3, HS I

Exercises: Friday, 14-16, E2 4, SR5 - Room 215


Exams:     End-term:    Re-exam:


  • 50% of the points in the exercises are needed to take part in the exams.
  • An exam is passed if you get at least 50% of the points.
  • The grading is based on the better result of the end-term and re-exam.
  • Exams can be oral or written (depends on the number of participants).


Prof. Dr. Matthias Hein

Office Hours: Mo, 16-18, Do, 16-18

Organization: to be announced