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Lecturer(s)
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Cvejn Jan, doc. Ing. Ph.D.
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Course content
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Introduction. Domains of use of optimization. Types of optimization problems. Parametrization. Mathematical apparatus for optimization - linear spaces, linear mappings, quadratic forms, differentiability, Taylor expansion of multiparameter functions. Mathematical formulation of optimization problem, types of extremes. Problems without constraints - necessary and sufficient conditions of the minimum. Minimum of a quadratic function, least-squares problem. Numerical solution with QR and SVD decompositions. Numerical algorithms for solving smooth problems without constraints. Division of the methods. Rate of convergence. Methods not using function model. Methods using direction search. Steepest descent method, Newton metod. Quasi-Newtonon methods and conjugate-gradient methods. Restricted step principle. Levenberg-Marquardt methods. Non-linear least squares problem and solving systems of non-linear equations. Problems with linear constraints of equality type. Linear programming. Elementary problems. Standard LP form. Simplex method. Basics of theory of optimization with constraints. Necessary and sufficient conditions. Convex problems. Duality. Principles of numerical solving problems with equality-type constraits. Penalty functions, method of extended Lagrangian. Elimination of variables. Lagrange-Newton method. Solving problems with constraints of equality and inequality type. Barrier functions. Active set method. Quadratic programming, Sequential quadratic programming. Projection methods. Interior point methods. Integer optimization problems. Branch and bound method. Approaches to global optimization problem. Methods using local optimization from randomly generated points. Covering methods. Methods of generalized local search. Random search methods. Evolution methods. Introduction into the theory of continuous optimal processes. Formulation of the optimal control problem. Necessary and sufficient conditions of optimality. Control variable contraints. Pontryagin maximum principle.
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Learning activities and teaching methods
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Monologic (reading, lecture, briefing), Work with text (with textbook, with book), Methods of individual activities
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Learning outcomes
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The goal of the subject is providing introduction into theoretical apparatus of methods of optimization and making an overview of the most important algorithms of searching for the optimum. An emphasis is posed on solving practical problems, which are chosen so that they demonstrate properties of particular methods. In the end of the semester approaches the global optimization problem are discussed. For solving problems utilization of Matlab software is assumed.
Obtaining an introduction into theoretical aparatus of optimization methods and providing an overview of the most important algorithms of searching optimum.
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Prerequisites
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Knowledge of differential and integral calculus, linear algebra.
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Assessment methods and criteria
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Oral examination, Written examination, Work-related product analysis
Credit, oral examination.
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Recommended literature
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Alexejev, V. M., Tichomirov, V. M., Fomin, S. V. Matematická teorie optimálních procesů, Praha, 1991..
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Bryson, A.E., Ho, Y.C. Applied Optimal Control. Hemisphere Corp., 1981.
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Fletcher, R. Practical Methods of Optimization. John Wiley & Sons Ltd., 2nd edition, 1987.
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Maňas, M. Optimalizační metody. SNTL, Praha, 1979.
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Nocedal, J., Wright, S. J. Numerical optimization. Springer Verlag, 1999.
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Stengel, R. Optimal Control and Estimation. Dover Publications, 1994.
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Štecha, J. Optimální rozhodování a řízení.Praha, ČVUT, 2000. Praha: ČVUT, 2004. ISBN 80-01-03010-5.
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