### Course: Optimization Methods

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 Course title Optimization Methods KMD/OPM Lecture + Lesson Master not specified Winter 5 Czech Compulsory Face-to-face Course does not contain work placement None The course is available to visiting students
Lecturer(s)
• Knobloch Roman, Mgr. Ph.D.
• Soudský Filip, RNDr. Ph.D.
Course content
Lectures: 1. Motivational problems 2. Aproximational methods for one-dimensional problems - Taylor polynomial, Newton tangential method 3. Introduction to Banach and Hilbert spaces 4. Function spaces $\mathcal{C}^k$ and spaces of Holder continuous functions 5. Fixed point theorems and their applications - proof of Picard's theorem 6. Approximative solution of differential equations 7. Dynamical systems, stability of solution 8. Dynamical systems, optimal regulation 9. Introduction to game theory, pure and mixed strategies, games with full and partial information 10. More on fixed point theorems (Brouwer's theorem) 11. Nash's theorem on existence of equilibrium point. 12. Some examples continuous games and existence of optimal strategy 13. Pursuit games 14. Some applications of game theory in problems fo economy biology and other disciplines Exercises: Practice topics according to lectures.

Learning activities and teaching methods
Monological explanation (lecture, presentation,briefing), Written assignment presentation and defence
• Class attendance - 56 hours per semester
• Preparation for credit - 15 hours per semester
• Semestral paper - 20 hours per semester
Learning outcomes
The subject focuses on solution of optimalization problems by means of classical and modern methods of optimalization. The initial part is devoted to numerical solutions of certain problems, for which the analytical methods fail. Namely finding the root of equation f(x)=0, approximative evaluation of functions values and approximative solution of ordinary differential equations. In the following part we shall give a brief introduction to game theory, we shall recall the classical result on existence of equilibria of the game (Nash theorem) and we will discuss the methods of finding the optimal strategy of the game. In the end we apply the theory in solving several problems motivated in biology and economy.
Knowledge of fundamentals of numerical mathematics.
Prerequisites
Passing of mathematical lectures of first four semestrs.

Assessment methods and criteria
Oral exam, Written exam

Credit: Working out a semestral work. Exam: Written.
Recommended literature
• Border, Kim C. Fixed point theorems with applications to economics and game theory. Cambridge university press, 1985. ISBN 9780521388085.
• Hans Peters. Game Theory A Multi-Leveled Approach. 2008. ISBN 978-3-540-69290-4.
• Martin Chvoj. Pokročilá teorie her. 2013. ISBN 9788024746203.
• Petr Habala, Petr Hájek, Václav Ziezler. Introduction to Banach spaces. Matfyzpress, 1996. ISBN 8085863146.
• Rufus Isac. Differential Games. New York-John Willey and sons, 1964. ISBN 0486406822.
• Simon, D.:. Evolutionary Optimization Algorithms. New Jersey, John Wiley, 2013. ISBN 978-0-470-93741-9.

 Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester