Course detail

Optimization Methods II

FSI-VPPAcad. year: 2024/2025

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. Project management, CMP and PERT methods. Algorithms for shortest paths in graphs and the branch and bound method. 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. Applications of the methods in solving practical problems.

Language of instruction

Czech

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.
Bertsekas, D. P.: Dynamic Programming and Optimal Control: Vol. I. Athena Scientific, Nashua. 2017.
Brucker, P.: Scheduling Algorithms. Springer-Verlag, Berlin, 2010.
Puterman, M. L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley-Interscience, New Jersey, 2005.

Recommended reading

Ahuja, R. K.; Magnanti, T. L.; Orlin, J. B.: Network Flows. Prentice Hall, Upper Saddle River, New Jersey, 1993.
Bertsekas, D. P.: Dynamic Programming and Optimal Control: Vol. II: Approximate Dynamic Programming. Athena Scientific, Nashua. 2012.
Boyd, S; Vandenberghe, L.: Convex Optimization. Cambridge University Press, 2004.
Kerzner, H.: Project Management: A Systems Approach to Planning, Scheduling, and Controlling. Wiley, New Jersey, 2009.
Klapka, J.; Dvořák, J.; Popela, P.: Metody operačního výzkumu. VUTIUM, Brno, 2001.
Pinedo, M. L.: Scheduling: Theory, Algorithms, and Systems. Springer-Verlag, Cham, 2016.
Volek, J; Linda, B.: Teorie grafů - aplikace v dopravě a veřejné správě. Univerzita Pardubice, 2012.
Winston W.L.: Operations Research. Applications and Algorithms. Thomson - Brooks/Cole, Belmont 2004.

Elearning

eLearning: currently opened course

Classification of course in study plans

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

  • Programme C-AKR-P Lifelong learning

    specialization CZS , 1 year of study, winter semester, elective

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. Basics of network analysis, topological ordering, CPM.
6. Calculation by stochastic evaluation of activities (PERT method).
7. Algorithms for shortest paths in a graph, the branch and bound method.
8. Multicriteria and constrained optimal control problems.
9. Deterministic continuous time optimal control, Hamilton-Jacobi-Bellman equation, Pontryagin maximum principle.
10. LQR a Kalman filter.
11. Problems with an infinite number of stages.
12. Process scheduling.
13. Approximate dynamic programming and Model predictive control.

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

Ing. David Ibehej

Syllabus

1. Solving dynamic programming problems in Matlab.
2. Resource allocation problems.
3. Dynamic programming in stochastic processes, optimizing a repair schedule.
4. State augmentation, optimal inventory control.
5. Viterbi algorithm, decoding of convolutional codes.
6. Examples of project graphs and networks. Implementation of CPM.
7. Numerical applications of the PERT method.
8. Algorithms for shortest paths in a graph, implementation of the A* algorithm.
9. Implementation of the branch and bound method.
10. Multicriteria knapsack problem.
11. LQR, drone control.
12. Job scheduling.
13. Semestral projects.

Elearning

eLearning: currently opened course