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Chapters on modern optimization and equilibria - NMEK606

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:	summer
E-Credits:	3
Hours per week, examination:	summer 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.
Teacher(s):	RNDr. Michal Červinka, Ph.D. doc. RNDr. Petr Lachout, CSc. doc. Ing. Jiří Outrata, DrSc.
Class:	Pravděp. a statistika, ekonometrie a fin. mat.
Classification:	Mathematics > Optimization

Opinion survey results Examination dates SS schedule Noticeboard

Annotation -

The lecture builds up base of modern optimization and equilibria theory.

Last update: T_KPMS (09.05.2014)

Aim of the course -

(i)

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.

(ii)

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 -

Lecture.

Last update: T_KPMS (09.05.2014)

Requirements to the exam -

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Requirements to exam

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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.

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Alternative requirements to exam in crisis situation

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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: Lachout Petr, doc. RNDr., CSc. (29.04.2020)

Syllabus -

Nonconvex nonsmooth analysis

1) Variational geometry nonconvex sets. (Several types of normal cones and their relations).

2) Subdiferentials and coderivatives (Fréchet, proximal, Clarke, Mordukhovich).

3) First order calculus with relaxed constraint qualification based on chain rules.

4) Second order calculus (derivatives of compositions for polyhedral and conic constraint systems).

5) Applications: Optimality conditions, stability analysis of multifunction, error bound property, nonsmooth numerical methods.

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)