Saarland University, Machine Learning Group, Fak. MI - Mathematik und Informatik, Campus E1 1, 66123 Saarbrücken, Germany

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

# TEACHING

## CONVEX OPTIMIZATION

Sommersemester 2014

### RESULT OF RE-EXAM and FINAL GRADES

Exam Inspection will be on Wednesday, 22th of October, 15.00-16.00 in Room 222.2

### 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.

Type: Advanced course (Vertiefungsvorlesung), 9 credit points

### LECTURE MATERIAL

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 AND EXCERCISES

 17.04. Introduction/Quick review LA and Analysis 22.04. Convex Sets Exercise 1 Solution 1 24.04. Convex sets (cont.) + Convex Functions 29.04. Convex Functions II Exercise 2 Solution 2 01.05 - Public Holiday 06.05. Convex Functions III (Subdifferential) Exercise 3 Solution 3 08.05. Convex Functions IV (Sublinear Functions) 13.05. Convex Functions V (Conjugate) Exercise 4 Solution 4 20.05. Convex Optimization Problems Exercise 5 Solution 5 22.05. Duality Theory 26.05. KKT conditions Exercise 6 Solution 6 28.05. - Public Holiday 03.06. One-dimensional Convex Optimization Exercise 7 Solution 7 05.06. Unconstrained Optimization 10.06. Unconstrained Optimization II Exercise 8 Solution 8 12.06. Lecture canceled 17.06. Subgradient Methods/Constrained Newton Exercise 9 Solution 9 19.06. Public Holiday 24.06. Interior Point Method Exercise 10 Data Solution 10 26.06. Projected Gradient Descent 01.07. Lecture cancelled Exercise 11 Data from Exercise 10 Solution 11 03.07. Accelerated First Oder Methods 08.07. Primal Dual First Order Methods Exercise 12 Solution 12 10.07. Coordinate Descent 15.07. Randomized Coordinate Descent 17.07. RCD/Efficient Implementation 22.07. Stochastic Gradient Descent

### TIME AND LOCATION

Lecture: Tuesday, 10-12, E2 4, SR6 - Room 217, Thursday, 10-12, E2 4, SR6 - Room 217

Exercises: Thursday, 8-10, E1 3, SR 16

End-term: 1.8., 14-17, HS 2 in E1 3, Re-exam: 10.10, 14-17, HS 2 in E1 3

• 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: Do, 16-18

Organization: to be announced

### 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

### NEWS

Check of first exam on October 7th, 11.00-12.00 in Room 222.2

Exam dates posted

Update of exercise sheet 10: added missing factor 1/2 in the definition of phi

Update of exercise sheet 10: The values for the error parameter C which are supposed to be used in the last part of the exercise have been added

03.05. - Registration is now possible in HISPOS until 19th of May.