Teaching

The WG Numerical Optimization provides the following lectures in the field of optimization:

Optimization I (Unrestricted Nonlinear Optimization):

The course consists of a two-hour lecture and a one-hour tutorial and is offered every summer semester. The course provides an introduction to the research field of numerical optimization. Unconstrained optimization problems are considered and different numerical solution methods are introduced. An outline of the lecture is as follows:

  1. Introduction
  2. Optimality Conditions
  3. Convexity and Convex Optimization
  4. Descent Methods and Step Size Strategies – Part 1
  5. Descent Methods and Step Size Strategies – Part 2
  6. Descent Methods and Step Size Strategies – Part 3
  7. Steepest descent and conjugate gradient methods - Part 1
  8. Steepest descent and conjugate gradient methods - Part 2
  9. Rates of Convergence
  10. Newton’s Method - Part 1
  11. Newton’s Method - Part 2
  12. Quasi-Newton Methods

Prerequisites for the course: Analysis I and II, Linear Algebra, Numerics I, PYTHON programming knowledge.

Creditability:
- Bachelor Mathematics: Supplementary module
- Bachelor Mathematical Finance: Module Numerics and Optimization
- Master of Education Mathematics: Elective module and oral final examination

Optimization II (Restricted Nonlinear Optimization):

This course is a four-hour lecture and a two-hour tutorial. The course is offered in the winter term and focuses on constrained nonlinear optimization. An outline of the lecture is as follows:

  1. Optimality Conditions for Constrained Optimization - Part 1
  2. Optimality Conditions for Constrained Optimization - Part 2
  3. Introduction to Linear Programming
  4. Interior-Point Methods for Linear Programming – Part 1
  5. Interior-Point Methods for Linear Programming – Part 2
  6. Quadratic Programming – Part 1
  7. Quadratic Programming – Part 2
  8. Penalty Methods
  9. Augmented Lagrangian Method
  10. Nonlinear Problems with Box Constraints – Part 1
  11. Nonlinear Problems with Box Constraints – Part 2
  12. Sequential Quadratic Programming – Part 1
  13. Sequential Quadratic Programming – Part 2

The lecture is divided into two parts, of which in particular only the first part (4.5 ECTS) can be attended separately.

Prerequisites for the course: Analysis I and II, Linear Algebra, Numerics I, Optimization I, PYTHON programming skills.

Creditability:
- Master Mathematics: Main and elective module
- Master Mathematical Finance: Elective module
- Master of Education Mathematics: Elective module and oral final exam (e.g. the first part of Optimization II together with Optimization I)

Mathematical Optimization for Data Analysis:

Optimization problems and methods play a central role in data science and machine learning. The scope of this lecture is to introduce the necessary mathematical basics for the use of specific optimization techniques in data analysis. The course is designed for students which are interested in a mathematical approach to this topic. Therefore, also students from mathematics and financial mathematics, but also from other departments (e.g. Department of Computer and Information Science or Department of Physics) are very welcome.

Hint: The present lecture is designed in coordination with Jun.-Prof. Dr. Tobias Sutter (Department of Computer and Information Science of the University of Konstanz). Thus, an additional participation in the lecture Optimization for Data Science is very welcome.

Table of Contents: The lecture contains the following topics

  1. Introduction
  2. Gradient Method Using Momentum
  3. Stochastic Gradient#
  4. First-Order Methods for Constrained Optimization
  5. Nonsmooth Functions and Subgradients
  6. Nonsmooth Optimization Methods
  7. Duality and Algorithms

Required knowledges: The lectures is essentially based on the lecture Optimization I. The successful participation in the lecture Optimization II is recommended, but not absolutely necessary. Knowledges in analysis (Analysis I & II), linear algebra (Linear Algebra I & II) and basics in measure theory (e.g., Analysis III) and stochastics (e.g., Stochastic Processes) are expected.

ECTS points: 5 (2h lecture plus 1h exercises per week)

Type of exam: oral or written exam (depending on the number of students)

Creditability:
- Master Mathematics: Specialization module
- Master Financial Mathematics: Elective module

Further Specialization in Optimization:

For further specialization in numerical optimization, lectures from the following list are offered on a regular basis (in addition to specialized seminars):

- Optimization of Elliptic Differential Equations (2 h Lecture, 1 h Tutorial, 5 ECTS)
- Optimal Control of Differential Equations (2 h Lecture, 1 h Tutorial, 5 ECTS)
- Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2 h Lecture, 1 h Tutorial, 5 ECTS)

Creditability:
- Master Mathematics: Specialization module
- Master Mathematical Finance: Elective module

Previous lectures

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