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Course, academic year 2019/2020
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Chapters on modern optimization and equilibria - NMEK605
Title in English: Kapitoly z moderní optimalizace a ekvilibrií
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019 to 2019
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: not taught
Language: Czech
Teaching methods: full-time
Note: you can enroll for the course repeatedly
Guarantor: RNDr. Michal Červinka, Ph.D.
doc. RNDr. Petr Lachout, CSc.
doc. Ing. Jiří Outrata, DrSc.
Class: Pravděp. a statistika, ekonometrie a fin. mat.
M Mgr. PMSE > Volitelné
Classification: Mathematics > Optimization
Annotation -
Last update: T_KPMS (09.05.2014)
The lecture builds up base of modern optimization and equilibria theory.
Aim of the course -
Last update: T_KPMS (25.04.2016)


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.

Course completion requirements -
Last update: doc. RNDr. Petr Lachout, CSc. (11.10.2017)

The course is finalized by exam.

Literature - Czech
Last update: T_KPMS (09.05.2014)

[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.

Teaching methods -
Last update: T_KPMS (09.05.2014)


Requirements to the exam -
Last update: doc. RNDr. Petr Lachout, CSc. (11.10.2017)

The exam is oral.

Examination is checking knowledge of all matters read by the course lecturer.

Syllabus -
Last update: T_KPMS (09.05.2014)

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

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

basic of optimization theory, convex analysis

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