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Course, academic year 2023/2024
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Chapters on modern optimization and equilibria - NMEK605
Title: Kapitoly z moderní optimalizace a ekvilibrií
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021 to 2023
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Note: you can enroll for the course repeatedly
Guarantor: doc. RNDr. Petr Lachout, CSc.
RNDr. Michal Červinka, Ph.D.
doc. Ing. Jiří Outrata, DrSc.
Class: Pravděp. a statistika, ekonometrie a fin. mat.
Classification: Mathematics > Optimization
Annotation -
The lecture builds up base of modern optimization and equilibria theory.
Last update: T_KPMS (09.05.2014)
Aim of the course -


Lecture builds up fundaments of variation geometry and of calculus for nonsmooth singlevalued and multivalued mappings. The main task is to develop the generalized differential calculus of the first and the second order, variational principles and stability theory of multivalued mappings.


The theory is applied to particular problems of optimization and game theory. The considered problems belong to generalized problems of mathematical programming, variational and quasi-variational inequalities, noncooperative equilibria and games with hierarchic structure.

Last update: T_KPMS (25.04.2016)
Course completion requirements -

The course is finalized by exam.

Last update: Lachout Petr, doc. RNDr., CSc. (11.10.2017)
Literature - Czech

[1] B.S. Mordukhovich: Variational Analysis and Generalized Differentiation, Vol. 1: Basic Theory, Vol. 2: Applications, Springer, Berlin, 2006.

[2] R. T. Rockafellar: Applications of convex variational analysis to Nash equilibrium, Proceedings of 7th International Conference on Nonlinear Analysis and Convex Analysis (Busan, Korea, 2011), 173-183.

[3] R.T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin 1998.

[4] W. Schirotzek: Nonsmooth Analysis, Springer, Berlin, 2007.

Last update: T_KPMS (09.05.2014)
Teaching methods -


Last update: T_KPMS (09.05.2014)
Requirements to the exam -


Requirements to exam


The exam is oral.

Examination is checking knowledge of all topics read at the lecture and parts given to self-study by the course lecturer.


Alternative requirements to exam in crisis situation


The exam is oral and will be organized either in a presence form or in distance online form.

Examination is checking knowledge of all topics specified by the course lecturer.

Last update: Červinka Michal, RNDr., Ph.D. (13.05.2020)
Syllabus -

Nonsmooth convex analysis in finite dimension

1) Summary on convex sets and functions; Lipschitz continuity of functions; semicontinuity of functions

2) Modern version of convex separation theorems; extremal systems of sets

3) Geometry of convex sets: convex tangent and normal cones; convex calculus; basic properties of multifunctions

4) Convex subdifferential; calculus; support functions

5) Duality; Fenchel conjugates

6) Convex nonsmooth optimization problems: applications and source problems; existence of a solution; optimality conditions and constraint qualification (Slater CQ, LICQ, MFCQ, calmness CQ, Abadie CQ, Guignard CQ); duality in convex programming, selected subgradient methods

7) Nash games (NEP) and equilibria: applications and source problems; existence of a solution

Last update: T_KPMS (09.05.2014)
Entry requirements -

basic of optimization theory, convex analysis

Last update: Lachout Petr, doc. RNDr., CSc. (30.05.2018)
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