Course objectives:
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A goal of the subject is to apprise the students of theoretical basis of the methods of optimization and of the most important numerical minimum-search algorithms in continuous domain. Practical verification of the explained methods by solving technical problems is stressed.
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Requirements on student
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Grant of credits is subject to presence on exercises and working out all submitted tasks. The examination has written and oral part.
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Content
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Topics of lectures by weeks:
1. Introduction, application areas of optimization. Types of optimization problems. Parametrization.
2. Mathematical apparatus for optimization: linear spaces, linear mapping, quadratic forms, differentiability, Taylor expansion of multivariable functions.
3. Mathematical formulation of optimization problem, kinds of extremes. Problems without constraints - necessary and sufficient conditions of minimum. The minimum of quadratic function. Least-squares problem.
4. Numerical algorithms for solving smooth problems without constraints - the subdivision. The rate of convergence. Methods not using the function model. The method of flexible simplex.
5. Global convergence. Direction search methods. Steepest descent method.
6. Newton´s method. Restricted step principle. Levenberg-Marquardt methods.
7. Quasi-Newton and conjugate-gradient methods.
8. The problems with linear equality contraints - solving by elimination nad orthogonal factorization.
9. Linear programming. Elementary problem types. Standard form of LP. Simplex method.
10. Basics of the theory of constrained optimization - necessary conditions of the minimum. Convex problems.
11. Principles of numerical treatment of the problems with non-linear equality constraints. Penalty functions. The principle of constraint linearization. Lagrange-Newton method.
12. Numerical treatment of the problems with non-linear constraints of both equality and inequality type. Utilization of barrier functions, the interior point methods. Active set method, quadratic programming, sequential quadratic programming.
13. Approaches to the global optimization problem. Methods utilizing local optimization from random-generated points. Covering methods. Generalized local search methods. Random search methods.
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Activities
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Fields of study
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CVEJN, J. Metody optimalizace. Elektronický studijní materiál.
V případě mimořádných opatření bude výuka probíhat vzdáleně s využitím programu MS Teams v době dle rozvrhu. Účast na schůzkách skupiny v MS Teams je ekvivalentní účasti na přednáškách a cvičeních.
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Guarantors and lecturers
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Literature
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Recommended:
NOCEDAL, J., WRIGHT, S. J. Numerical optimization. 2nd edition.. 2006.
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Recommended:
FLETCHER, R. Practical Methods of Optimization. 2nd edition.. 1987.
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Praktická výuka
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52
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Domácí příprava na výuku
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32
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Příprava na zkoušku
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40
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Semestrální práce
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26
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Total
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150
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Prerequisites - other information about course preconditions |
Mathematics - diferential and integral calculus, matrices. MATLAB programming. |
Competences acquired |
Student after passing the subject:
- proves teoretical knowledge
- is able to implement algorithms for minimization of multi-dimmensional functions, both without constraints and with constraints.
- is able to utilize programmed algorithms, as well as the algorithms built in the software MATLAB, for solving technical problems.
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Teaching methods |
- Monologic (reading, lecture, briefing)
- Methods of individual activities
- Demonstration
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Assessment methods |
- Oral examination
- Written examination
- Home assignment evaluation
- Student performance assessment
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