Lecturer(s)


Burkotová Jana, Mgr. Ph.D.

Andrášik Richard, Mgr.

Netuka Horymír, RNDr. Ph.D.

Machalová Jitka, RNDr. Ph.D.

Course content

1. Unconstrained optimization  subject, applications, examples. Introductory definitions and basic conceptions. 2. Firstorder necessary optimality conditions. Secondorder necessary optimality conditions. Secondorder sufficient optimality conditions. 3. Univariate minimization. Minimization without using derivatives (comparative method, Fibonacci search method, golden section search method). Methods using derivatives (bisection, Newton method). 4. Derivativefree minimization of functions of several variables. NelderMead method. HookeJeeves method. 5. Minimization of quadratic functions using gradient methods  part I. Quadratic function and its properties. Descent methods basics. Method of steepest descent for quadratic functions. 6. Minimization of quadratic functions using gradient methods  part II. Conjugate gradient method. Convergence analysis. 7. Line search methods  part I. Fundamental principles. Step length selection using backtracking line search. Armijo condition. BacktrackingArmijo line search algorithm. Convergence analysis. 8. Line search methods  part II. Wolfe conditions. Inexact line search using Wolfe conditions. Convergence analysis. 9. Line search methods  part III. Method of steepest descent for nonquadratic function. Conjugate gradient method for nonquadratic function and its two main versions. 10. Newton's method and its modifications. Classical Newton method. Modifications of Newton's method (damped Newton's method, finitedifference Newton's method). 11. QuasiNewton methods. Principles of quasiNewton methods. General scheme with B matrices. General scheme with G matrices. Broyden's method, DFP method, BFGS method. 12. Solution of systems of nonlinear equations. Multivariate minimization and its connection with nonlinear algebraic equations. Newton's method. Broyden's method.

Learning activities and teaching methods

Lecture, Monologic Lecture(Interpretation, Training), Demonstration
 Attendace
 39 hours per semester
 Homework for Teaching
 15 hours per semester
 Preparation for the Exam
 40 hours per semester

Learning outcomes

Gain knowledge about theory and algorithms required to solve unconstrained optimization problems.
Knowledge Gain useful knowledge about theory and algorithms in order to study and solve unconstrained optimization problems.

Prerequisites

Standard knowledge from mathematical analysis and linear algebra. Information concerning numerical methods are welcomed, but not necessary. Elemental experience with computation on PC.
KMA/NM1

Assessment methods and criteria

Oral exam, Seminar Work
Credit: the student has to compute given examples. Exam: the student has to understand the subject and be acquainted with theory and computational methods.

Recommended literature


J. E. Dennis, R. B. Schnabel. (1983). Numerical Methods for Unconstrained Optimization and Nonlinear Equations. PrenticeHall Englewood Cliffs N. J.

J. Machalová, H. Netuka. (2013). Numerické metody nepodmíněné optimalizace. Olomouc.

J. Nocedal, S. J. Wright. (1999). Numerical Optimization. Springer.

L. Lukšan. (2011). Numerické optimalizační metody. Nepodmíněná minimalizace. Technical report no. 1152. Praha.

M.S. Bazaraa, H.D. Sherali, C.M. Shetty. (2006). Nonlinear Programming. Theory And Algorithms..

O. Došlý. (2005). Základy konvexní analýzy a optimalizace v R^n. Brno.

S. Míka. (1997). Matematická optimalizace. FAV ZČU, Plzeň.

Z. Dostál, P. Beremlijski. (2012). Metody optimalizace. Ostrava.
