Data Science and Artificial Intelligence

Optimization and Numerics

Integrated course, 2.50 ECTS

 

Course content

Part 1: Aspects of numerics
- Numerical presentation on the computer
- Type and reduction of numerical errors
- Conditioning problems
- Numerical differentiation and numerical quadrature
- Numerical solving of systems of equations (including Newton's method)
- Pivoting and matrix decomposition (LU, QR, ...)
Part 2: Optimization
- Basic aspects of optimization tasks
- One- and multi-dimensional extreme value tasks
- 1st order descent procedure (steepest descent, impulse methods, ...)
- 2nd order descent procedure (Newton and Newton-style procedures, ...)
- Conjugate gradients
- Linear optimization, simplex algorithm, MILP problems
- Optimization with constraints (long-range approach including KKT conditions)
- Multi-criteria optimization (including Pareto analysis)
- Special methods of stochastic optimization (e.g. simulated anealing)

Learning outcomes

The students have a profound understanding of the special features of numerical methods for solving mathematical problems. They can identify numerical problems and minimize numerical errors. They have a profound understanding of optimization tasks and can assess which method is suitable for which optimization problem and then apply this method.

Recommended or required reading and other learning resources / tools

Recommended literature or books:
- Arens, T. et al. (2018). Mathematics, Springer spectrum, 4th edition
- Dahmen, W., Reusken, A. (2012): Numerik für Ingenieure und Naturwissenschaftler, Springer, 2nd edition
- Hart, W.E. et al. (2017). Pyomo - Optimization Modeling in Python, Springer, 2nd ed.
- Knuth, D.E. (2011) The Art of Computer Programming (4 volumes), Addison Wesley, Revised Edition
- Papageorgiou, M., Leibold, M., Buss, M. (2015). Optimization - static, dynamic, stochastic methods for use, Springer-Vieweg, 4th edition
- Sedgewick, R., Wayne, K. (2014). Algorithms: algorithms and data structures, Pearson study, 4th edition
- Skiena, S. S. (2012). The Algorithm Design Manual, Springer, corr. reprint
- Weicker, K., Weicker, N. (2013). Algorithmen und Datenstrukturen, Springer Vieweg
- Weitz, E. (2018). Concrete mathematics (not only) for computer scientists: With many graphics and algorithms in Python. Springer spectrum: 1st edition.
Recommended journals or selected articles:
Relevant journals and articles will be announced in the courses.

Typical software for this module:
Python / Spyder / PyCharm, Matlab / Octave / Scilab etc.

Mode of delivery

1,25 ECTS Lecture, 1,25 ECTS Exercise

Prerequisites and co-requisites

Module 5

Assessment methods and criteria

Lecture: final exam, Exercise: immanent test character