Linear and Convex Optimization

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

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. In particular, they understand duality concepts, can derive optimality conditions and understand optimization algorithm for linear optimization problems, i.e the dual simplex algorithm.

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
Bachelor 2019: MA0001 Analysis 1, MA0002 Analysis 2, MA0004 Lineare Algebra und Diskrete Strukturen 1, MA0005 Lineare Algebra 2 und Diskrete Strukturen

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.

blackboard, exercise sheets

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.

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.