Lecturer(s)


Netuka Horymír, RNDr. Ph.D.

Pastor Karel, doc. Mgr. Ph.D.

Burkotová Jana, Mgr. Ph.D.

Course content

1. Constrained optimization, its importance for applications, examples. Introductory definitions and basic conceptions. 2. Firstorder necessary optimality conditions. KarushKuhnTucker conditions. Geometric interpretation of KKT conditions. 3. Firstorder optimality conditions for minimization on convex sets. Sufficient optimality conditions for problems of this type. 4. Constraint qualifications in nonlinear programming problems. Useful constraint qualifications conditions. 5. Lagrangian function. Secondorder necessary optimality conditions. Secondorder sufficient optimality conditions. 6. Saddle points of Lagrangian function and their connection with optimization problems. Lagrangian dual problems and their properties. 7. Complementarity problems and their relation to nonlinear programming. Linear complementarity problem. Lemke's method. 8. Quadratic programming and its importance. Methods for solution of problems with equality constraints. 9. Active set method for convex quadratic programming problems having inequality constraints. 10. Methods for nonlinear programming problems with linear constraints  nullspace method and gradient projection method. 11. Penalty methods for general nonlinear programming. Quadratic penalty function, barrier functions. Augmented Lagrangian method. 12. Principles of sequential quadratic programming method. Concept of interiorpoint methods.

Learning activities and teaching methods

Lecture, Monologic Lecture(Interpretation, Training), Demonstration
 Attendace
 52 hours per semester
 Homework for Teaching
 25 hours per semester
 Preparation for the Exam
 45 hours per semester

Learning outcomes

Gain knowledge about theory and algorithms required to solve nonlinear programming problems.
Knowledge Gain useful knowledge about theory and algorithms in order to study and solve nonlinear programs.

Prerequisites

Student has to pass the course Numerical methods of optimization. Standard knowledge from mathematical analysis and linear algebra. Information concerning numerical methods are welcomed, but not necessary. Elemental experience with computation on PC.
KMA/NMO

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 theoretical and practical aspects of the methods.

Recommended literature


D.G. Luenberger, Y. Ye. (2008). Linear And Nonlinear Programming. 3rd Edition.

J. Machalová, H. Netuka. (2013). Nelineární programování: Teorie a metody. Olomouc.

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

K.G. Murty. (1988). Linear Complementarity, Linear and Nonlinear Programming. Berlin.

M.S. Bazaraa, H.D. Sherali, C.M. Shetty. (2006). Nonlinear Programming. Theory And Algorithms. 3rd Edition.

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.
