TEACHING

GENERAL INFORMATION

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. We will also cover related non-convex problems such as d.c. (difference of convex) programming, biconvex optimization problems and hard combinatorial problems and their relaxations into convex problems. While the emphasis is given on mathematical and algorithmic foundations, several example applications together with their modeling as optimization problems will be discussed.

The course requires a good background in linear algebra and multivariate calculus, but no prior knowledge in optimization is required. The course 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.

The course counts as lecture in mathematics and computer science.

Type: Advanced course (Vertiefungsvorlesung), 9 credit points

LECTURE MATERIAL

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

Lecture notes (will be updated - coverage until convex sets): Lecture notes

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

SLIDES AND EXCERCISES

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20.04. - Introduction	Exercise 0	Solution 0
24.04. - Convex sets
27.04. - Convex Functions I	Exercise 1	Solution 1
01.05 - No lecture due to public holiday
04.05. - Convex functions + subdifferential	Exercise 2	Solution 2
08.05. - Normal cone/Optimality conditions
11.05. - Conjugate Function/Quasiconvex	Exercise 3	Solution 3
15.05. - Convex Optimization
18.05. - Duality Theory	Exercise 4	Solution 4
25.05. - No lecture due to public holiday
29.05. - Lecture canceled
01.06. - Sensitivity/KKT Conditions	Exercise 5	Solution 5
05.06. - No lecture due to public holiday
08.06. - Non-smooth KKT/One-dimensional convex Opt	Exercise 6	Solution 6
12.06. - Gradient Descent
15.06. - No lecture due to public holiday
19.06. - Newton and Quasi-Newton Method	Exercise 7	Solution 7
22.06. - Subgradient Descent
26.06. - Interior Point Method
29.06. - Interior Point Method (cont.)	Exercise 8	Solution 8	Material for Ex8
03.07. - Projected Grad/Subgrad Descent
06.07. - Accelerated Gradient Descent	Exercise 9	Solution 9
10.07. - Primal-Dual Proximal Method
13.07. - Lecture canceled	Exercise 10	Solution 10
17.07. - Primal-Dual Proximal Method II
20.07. - Coordinate Descent

TIME AND LOCATION

Lecture: Monday, 10-12, E1 3, HS 003, Thursday, 10-12, E1 3, HS 003

Tutorials:  Friday, 14-16, E1 3, SR 015

EXAMS AND GRADING

End-term: 11.08., E1 3, HS 001/002, 14-17, Re-exam: 09.10., E1 3, HS 001/002, 10-13

Grading:

• 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).

LECTURER

Prof. Dr. Matthias Hein

Office Hours: Thursday, 16-18

Organization: Pedro Mercado Lopez

LITERATURE AND OTHER RESOURCES

• D. P. Bertsekas: Convex Optimization Theory, (2009).
  Link to the free chapter on optimization algorithms.
• J.-B. Hiriart-Urruty, C. Lemaréchal: Fundamentals of Convex Analysis (2013).
• S. Boyd and L. Vandenberghe: Convex Optimization, Cambridge University Press, (2004).
  The book is freely available
• D. P. Bertsekas: Nonlinear Programming, Athena Scientific, (1999).
• Other resources:
  • Matlab is available on cip[101-114] and cip[220-238].studcs.uni-sb.de, gpool[01-27].studcs.uni-sb.de
    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 username@computername.studcs.uni-sb.de
  • Matlab tutorial by David F. Griffiths