SubjectsSubjects(version: 945)
Course, academic year 2023/2024
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Numerical Optimization Methods 1 - NMNV503
Title: Numerické metody optimalizace 1
Guaranteed by: Department of Numerical Mathematics (32-KNM)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Václav Kučera, Ph.D.
Class: M Mgr. MMIB > Povinně volitelné
M Mgr. MOD > Povinně volitelné
M Mgr. NVM > Povinné
Classification: Mathematics > Numerical Analysis
Incompatibility : {Předměty NMNV501 a NMNV534, místo nichž je nově NMNV503}
Interchangeability : {Předměty NMNV501 a NMNV534, místo nichž je nově NMNV503}
Is incompatible with: NMNV534, NMNV501
Is pre-requisite for: NMNV544
Is interchangeable with: NMNV534, NMNV501
Annotation -
Last update: doc. RNDr. Václav Kučera, Ph.D. (19.12.2018)
The course deals with the theoretical and practical questions of the numerical solution of non-linear equations and minimization of functionals. The first part is dedicated to the solution of nonlinear equations and their systems, we will focus mainly on Newton's method, its variants and modifications. The second part deals with the minimization of functionals, focusing on descent methods (e.g. the non-linear conjugate gradient method and quasi-Newtonian methods) and on trust region methods.
Literature -
Last update: doc. RNDr. Václav Kučera, Ph.D. (19.12.2018)

J. M. Ortega, W. C. Rheinboldt: Iterative solution of nonlinear equations in several variables. Academic Press new York and London, 1970.

C. T. Kelley: Solving Nonlinear Equations with Newton's Method. Philadelphia, SIAM 2003.

A. Ostrowski: Solution of Equations and Systems of Equations. Academic Press, New York 1960; second edition, 1966.

P. Henrici: Elements of Numerical Analysis. John Wiley and Sons, Inc. 1964.

P. Deufelhard: Newton Methods for Nonlinear Problems. Springer-Verlag Berlin Heidelberg, 2004.

R. Fletcher, Practical Methods of Optimization, 2nd edition Wiley 1987, (republished 2000).

D. G. Luenberger and Y. Ye, Linear and Nonlinear Programming, Third edition. Springer, New York, MA, 2008.

J. Nocedal and S. Wright, Numerical Optimization, Second edition, Springer Verlag 2006.

J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM 1996, originally published in 1983.

Syllabus -
Last update: doc. RNDr. Václav Kučera, Ph.D. (20.12.2018)

Basic numerical methods for solving scalar nonlinear equations (Newton's method and secant method), local convergence, order of convergence. More advanced methods (Muller's method, inverse quadratic interpolation, Brent's method).

Solution of systems of nonlinear equations, Newton's method, quasi-Newtonian methods. Global convergence, continuation methods.

Theory of unconstrained optimization (necessary and sufficient conditions, role of convexity).

Line search - the search for minima in the given descent direction (Goldstein, Armijo, Wolfe conditions). Basic descent methods (the method of steepest descent and the Newton method), conjugate direction methods (the nonlinear conjugate gradient method),

Quasi-Newton methods (rank-one update, DFP, BFGS, the Broyden family),

Trust-region methods, dogleg.

Least-squares problems (the Gauss-Newton and the Levenberg-Marquart method).

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