|
Lecturer(s)
|
-
Cvejn Jan, doc. Ing. Ph.D.
|
|
Course content
|
1. Introduction. Domains of use of optimization. Types of optimization problems. Parametrization. 2. Mathematical apparatus for optimization - linear spaces, linear mappings, quadratic forms, differentiability, Taylor expansion of multiparameter functions. 3. Mathematical formulation of optimization problem, types of extremes. Problems without constraints - necessary and sufficient conditions of the minimum. Minimum of a quadratic function. 4. Numerical algorithms for solving smooth problems without constraints. Division of the methods. Rate of convergence. Methods not using function model. Flexible simplex method. 5. Methods using direction search. Steepest descent method, Newton metod. Quasi-Newtonon methods and conjugate-gradient methods. 6. Restricted step principle. Levenberg-Marquardt methods. Problems with linear constraints of equality type. 7. Linear programming. Elementary problems. Standard LP form. Simplex method. 8. Basics of theory of optimization with constraints. Necessary and sufficient conditions. Convex problems. 9. Principles of numerical solving problems with equality-type constraits. Penalty functions, method of extended Lagrangian. Lagrange-Newton method. 10. Numerical solving problems with constraints of equality and inequality type. Barrier functions. Active set method. Quadratic programming, Sequential quadratic programming. 11. Approaches to global optimization problem. Methods using local optimization from randomly generated points. Covering methods. Methods of generalized local search. Random search methods. 12. Introduction into the theory of continuous optimal processes. Formulation of the optimal control problem. Necessary and sufficient conditions of optimality. 13. Control variable contraints. Pontryagin maximum principle. Principles of numerical solution of the optimal control problems.
|
|
Learning activities and teaching methods
|
|
Monologic (reading, lecture, briefing), Skills training, Work-related activities
|
|
Learning outcomes
|
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. The end of the semester is dedicated to the algorithms of global optimization and the introduction into the optimal control theory. For solving practical 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.
|
|
Prerequisites
|
Knowledge of differential and integral calculus, linear algebra.
|
|
Assessment methods and criteria
|
Oral examination, Written examination, Home assignment evaluation
Credit, oral examination.
|
|
Recommended literature
|
-
Alexejev, V. M., Tichomirov, V. M., Fomin, S. V. Matematická teorie optimálních procesů, Praha, 1991..
-
Bryson A. E., Ho Y.C. Applied Optimal Control . Hemisphere Corp., New York, 1981.
-
FLETCHER, R. Practical Methods of Optimization, 2nd edition. John Wiley & Sons, 1987.
-
Nocedal, J., Wright, S. J. Numerical optimization. Springer Verlag, 1999.
-
Stengel, R. Optimal Control and Estimation. Dover Publications, 1994.
-
ŠTĚCHA, J. Optimální rozhodování a řízení. Praha: ČVUT, 2004.
|