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 2016

### 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.

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

 21.04. - Introduction 26.04. - Convex sets Exercise 1 Solution 1 28.04. - Convex sets + Convex Functions 03.05. - Convex Functions II Exercise 2 Solution 2 05.05. - No lecture due to public holiday 10.05. - Convex Functions II Exercise 3 Solution 3 12.05. - Subdifferential 17.05. - Convex Optimization Exercise 4 Solution 4 19.05. - Convex Optimization 24.05. - Duality Theory Exercise 5 Solution 5 26.05. - No lecture due to public holiday 31.05. - KKT Conditions Exercise 6 Solution 6 02.06. - Sensitivity/non-smooth KKT conditions 07.06. - One-dimensional Opt/Grad Descent Exercise 7 Solution 7 Material for Ex7 09.06. - Newton Descent 14.06. - Subgradient Methods/Constrained Newton Exercise 8 Solution 8 Material for Ex8 16.06. - Barrier Method 21.06. - Barrier Method/Projected Grad Descent Exercise 9 Solution 9 Material for Ex9 23.06. - Projected Grad/Subgrad Descent 28.06. - Accelerated Gradient/Proximal Methods Exercise 10 Solution 10 01.07. - Primal-Dual Proximal Method 05.07. - Quasi Newton Exercise 11 Solution 11 07.07. - Coordinate Descent 12.07. - SDCA/Coordinate Gradient Descent Exercise 12 Solution 12 14.07. - Coordinate Gradient Descent 19.07. - Implementation/Parallelization 21.07. - Parallel Methods 26.07. - Stochastic Gradient Descent 28.07. - Nonconvex Optimization

### TIME AND LOCATION

Lecture: Tuesday, 14-16, E2 4, SR6 - Room 217, Thursday, 10-12, E1 3, HS 003

Tutorials:  Friday, 16-18, E2 4, SR6 - Room 217

End-term: 10.8.2016, 13-15 HS 001, Re-exam: 11.10.2016, 14-16 HS 003

• 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: Quynh Nguyen Ngoc

### 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:

### NEWS

Re-exam will be on October 11 at 14.00 in HS 003.

Results of the exam can be found here

Exercise 12 is the last exercise of this lecture.

The matlab file to minimize the softmax/cross entropy loss via Newton descent is in the material for Exercise 8.

List of students admitted to the exam: pdf