Course detail

Optimization Methods II

FSI-VPP-AAcad. year: 2025/2026

The course deals with the following topics: Dynamic programming and optimal control of stochastic processes. Bellman optimality principle as a tool for optimization of multistage processes with a general nonlinear criterion function. Optimum decision policy. Computational aspects of dynamic programming in discrete time. Hidden Markov models and the Viterbi algorithm. Algorithms for shortest paths in graphs. Multicriteria control problems. Deterministic optimal control in continuous time, Hamilton-Jacobi-Bellman equation, Pontryagin maximum principle. LQR and Kalman filter. Process scheduling and planning. Problems with infinitely many stages. Approximate dynamic programming. Heuristic methods for complex problems. Applications of the methods in solving practical problems.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. Ing. Jakub Kůdela, Ph.D.

Department

Institute of Automation and Computer Science (ÚAI)

Entry knowledge

Knowledge of the basics of programming, mathematical analysis, algebra, theory of sets, statistics and probability.

Rules for evaluation and completion of the course

Course-unit credit: Active participation in the seminars, elaboration of a given project. Examination: Written and oral.
Attendance at seminars is required. An absence can be compensated for via solving additional problems.

Aims

The aim of the course is to inform the students about creations and applications of mathematical methods for optimal control of technological and economic processes e.g. in the automation of mechanical systems, in the management of production in mechanical engineering, in project management and in optimization of information systems, using contemporary tools of computer science.

Knowledge: Students will know basic principles and algorithms of methods applicable to the optimization of the deterministic and stochastic, discrete and continuous. They will be made familiar with basic principles and algorithms of methods that are appropriate to creation of decision-support systems for project management, as the tool for the identification, selection and realization of projects. Skills: Students will be able to apply the above methods to the solution of the practical problems from economic decision, problems of increasing the reliability of technological devices, problems of automation control of technological processes and problems of project management, by using contemporary tools of computer science.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Bazaraa, M, S.; Sherali, H. D.; Shetty, C. M.: Nonlinear Programming. Wiley, 2013. (EN)
Bertsekas, D. P.: Dynamic Programming and Optimal Control: Vol. I. Athena Scientific, Nashua. 2017. (EN)
Brucker, P.: Scheduling Algorithms. Springer-Verlag, Berlin, 2010. (EN)
Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming. Springer, 2014. (EN)
Martí, R. Pardalos, P.M., Resende, M.G.C.: Handbook of Heuristics. Springer Cham, 2018. (EN)
Puterman, M. L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley-Interscience, New Jersey, 2005. (EN)

Recommended reading

Ahuja, R. K.; Magnanti, T. L.; Orlin, J. B.: Network Flows. Prentice Hall, Upper Saddle River, New Jersey, 1993. (EN)
Bertsekas, D. P.: Dynamic Programming and Optimal Control: Vol. II: Approximate Dynamic Programming. Athena Scientific, Nashua. 2012. (EN)
Boyd, S; Vandenberghe, L.: Convex Optimization. Cambridge University Press, 2004. (EN)
Kerzner, H.: Project Management: A Systems Approach to Planning, Scheduling, and Controlling. Wiley, New Jersey, 2009. (EN)
Pinedo, M. L.: Scheduling: Theory, Algorithms, and Systems. Springer-Verlag, Cham, 2016. (EN)
Winston W.L.: Operations Research. Applications and Algorithms. Thomson - Brooks/Cole, Belmont 2004. (EN)

Classification of course in study plans

  • Programme N-AIŘ-P Master's 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

doc. Ing. Jakub Kůdela, Ph.D.

Syllabus

1. Basics of mathematical processes theory. Bellman optimality principle and dynamic programming.
2. Minimax (robust) formulation. Reformulations and state augmentation.
3. Deterministic finite-state problems. Forward DP algorithm.
4. Hidden Markov models and the Viterbi algorithm.
5. Algorithms for shortest paths in a graph.
6. Multicriteria and constrained optimal control problems.
7. LQR a Kalman filter.
8. Problems with an infinite number of stages.
9. Deterministic continuous-time optimal control, Hamilton-Jacobi-Bellman equation, Pontryagin maximum principle.
10. Process scheduling.
11. Approximate dynamic programming.
12. Rolling horizon and Model predictive control.
13. Heuristics for complex problems - genetic algorithms and ant colony optimization.

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

doc. Ing. Jakub Kůdela, Ph.D.

Syllabus

The exercise follows the topics discussed in the lecture. The main focus is on the software implementation of the studied methods.