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Lecturer(s)
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Machalová Jitka, doc. RNDr. Ph.D., MBA
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Burkotová Jana, Mgr. Ph.D.
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Radová Jana, Mgr.
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Ženčák Pavel, RNDr. Ph.D.
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Course content
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1. Constrained optimization, its importance for applications, examples. Introductory definitions and basic conceptions. 2. First-order necessary optimality conditions. Karush-Kuhn-Tucker conditions. Geometric interpretation of KKT conditions. 3. First-order 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. Second-order necessary optimality conditions. Second-order 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 - null-space 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 interior-point methods.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Demonstration
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Learning outcomes
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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.
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Prerequisites
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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.
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Assessment methods and criteria
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Oral exam, Seminar Work
Credit: the student has to compute given examples. Exam: active participation, the student has to understand the subject and be acquainted with theoretical and practical aspects of the methods.
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Recommended literature
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J. Machalová, H. Netuka. (2013). Nelineární programování: teorie a metody. Olomouc: Univerzita Palackého v Olomouci, 165 s. Skripta.
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J. Nocedal, S. J. Wright. (2006). Numerical Optimization. Springer.
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L. Lukšan. (2011). Numerické optimalizační metody. Nepodmíněná minimalizace. Technical report no. 1152. Praha.
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M.S. Bazaraa, H.D. Sherali, C.M. Shetty. (2006). Nonlinear Programming. Theory And Algorithms.
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S. Míka. (1997). Matematická optimalizace. FAV ZČU, Plzeň.
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Z. Dostál, P. Beremlijski. (2012). Metody optimalizace. Ostrava.
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