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# Fundamentals of Convex Optimization

## Module MA2504

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of SS 2011

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2020/1SS 2020WS 2018/9WS 2011/2SS 2011

### Basic Information

MA2504 is a semester module in English language at Master’s level which is offered in summer semester.

This module description is valid from SS 2011 to SS 2018.

270 h 90 h 9 CP

### Content, Learning Outcome and Preconditions

#### Content

convex sets, convex functions, projection, separation, subdifferential, optimality conditions, polyhedra, linear optimization problems, duality, (dual) simplex algorithm, Karush-Kuhn-Tucker conditions, selected applications and further topics of convex analysis and linear optimization

#### Learning Outcome

After successful completion of the module students are able to understand and apply the basic notions, concepts, and methods of convex analysis and linear optimization. Moreover, they are familiar with the underlying geometry and can model problems arising in practice.

#### Preconditions

MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra and Discrete Structures 1, MA1102 Linear Algebra and Discrete Structures 2,
Helpful: MA2501 Algorithmic Discrete Mathematics, MA2503 Introduction to Nonlinear Optimization

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 4 Fundamentals of Convex Optimization [MA2504] Brandenberg, R. Mon, 16:00–18:00, virtuell
Tue, 12:00–14:00, virtuell
eLearning
UE 2 Fundamentals of Convex Optimization (Exercise Session) [MA2504] Brandenberg, R. Fiedler, M. dates in groups eLearning

#### Learning and Teaching Methods

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

#### Media

blackboard, exercise sheets

#### Literature

P. Gritzmann. Grundlagen der mathematischen Optimierung, Springer, 2013.
D. P. Bertsekas, A. Nedic, A. E. Ozdaglar. Convex Analysis and Optimization, Athena Scientific, 2003.
D. Bertsimas, J. N. Tsitsiklis. Introduction to Linear Optimization, Athena Scientific, 1997.
G. B. Dantzig, M. N. Thapa. Linear Programming 1: Introduction. Springer, 1997.
J.-B. Hiriart-Urruty, C. Lemarechal. Fundamentals of Convex Analysis, Springer, 2001.
C. H. Papadimitriou, K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Dover, 1998.
R. T. Rockafellar. Convex Analysis, Princeton University Press, 1970.
A. Schrijver. Theory of Linear and Integer Programming. Wiley, 1986.
R. J. Vanderbei. Linear Programming, Foundations and Extensions, Springer, 2008.

### Module Exam

#### Description of exams and course work

The module examination is based on a written exam (90 minutes). Students have to know the fundamental concepts and methods of convex analysis and of linear optimization and are familiar with the underlying geometry. They can adequately model problems in practical applications.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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