Course: Convex Analysis

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Course title Convex Analysis
Course code KMA/KAM
Organizational form of instruction Lecture + Exercise
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 4
Language of instruction Czech
Status of course Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Tomeček Jan, doc. RNDr. Ph.D.
  • Machalová Jitka, RNDr. Ph.D.
  • Pastor Karel, doc. Mgr. Ph.D.
Course content
1. Epigraphs of functionals. 2. Lower-semicontinuity and its characterization. 3. Convex sets. 4. Convex functionals. 5. Kernels and interiors of convex sets. 6. Continuity of convex functionals. 7. The principle of automatic continuity and the uniform boundedness principle. 8. The Hahn-Banach theorem. 9. Subdifferentials. 10. Normal cones. 11. Moreau-Rockafellar's theorems. 12. Unconstrained and constrained minimization of convex functionals. 13. Generalized convexity.

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
  • Attendace - 39 hours per semester
  • Homework for Teaching - 10 hours per semester
  • Preparation for the Course Credit - 20 hours per semester
  • Preparation for the Exam - 50 hours per semester
Learning outcomes
Understand convex functions, their subdifferentials and their usage in optimization.
Comprehension Understand the basic properties of convex functions and convex sets.
Prerequisites
Knowledge of differential calculus.

Assessment methods and criteria
Oral exam, Dialog

Credit: active participation, homework solving. Exam: the student has to understand the subject and be able to prove the principal results.
Recommended literature
  • J. Jahn. (2004). Vector Optimization: Theory, Applications and Extensions. Springer, Berlin.
  • N. Hadjisawas, S. Komlósi, S. Schaible. (2005). Handbook of generalized convexity and monotonicity. Springer New York.
  • V. M. Aleksejev, V. M. Tichomirov, S. V. Fomin. (1991). Matematická teorie optimálních procesů. Academia, Praha.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Mathematics and Applications (1) Mathematics courses 3 Winter