M5170 Mathematical Programming

Faculty of Science
Autumn 2022
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Roman Šimon Hilscher, DSc. (lecturer)
doc. Mgr. Petr Zemánek, Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Roman Šimon Hilscher, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 10:00–11:50 M2,01021
  • Timetable of Seminar Groups:
M5170/01: Wed 10:00–11:50 M4,01024, P. Zemánek
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
Students will get basic knowledge concerning mathematical programming, numerical methods of unconstrained optimization, and also convex analysis.
Learning outcomes
After passing the course, the student will be able:
(1) to define and interpret the basic notions used in the basic parts of convex analysis and to explain their mutual context;
(2) to formulate relevant mathematical theorems and statements and to explain methods of their proofs;
(3) to use effective techniques utilized in basic fields of convex analysis;
(4) to apply acquired pieces of knowledge for the solution of specific problems of convex programming and to some numerical methods of optimization including problems of applicative character.
Syllabus
  • I. Convex analysis: Convex sets (basic concepts, convex hull, separation and supporting hyperplanes); Convex Functions (basic concepts, criteria of convexity for differentiable functions); Subgradient and Subdifferential; Fenchel transformation; Systems of linear and convex inequalities.
  • II. Numerical methods of unconstrained minimization: One-dimensional minimization (brute-force search, dichotomous search, Fibonacci and golden ratio methods); Unconstrained optimization (Method of steepest descent, Newton method, Conjugate gradient method).
  • III. Mathematical programming: Lagrange principle (necessary and sufficient conditions for optimality, Kuhn-Tucker conditions, basic concepts of convex programming); Duality in mathematical programming (dual problem, Kuhn-Tucker vector, weak duality, strong duality, saddle point); Dependence on parameters (Envelope Theorem, shadow price)
Literature
    required literature
  • Petr Zemánek, Optimalizace aneb Když méně je více (učební text), viz https://optimalizace.page.link/ucebni_text
    recommended literature
  • DOŠLÝ, Ondřej. Základy konvexní analýzy a optimalizace v Rn. 1. vyd. Brno: Masarykova univerzita, 2005, viii, 185. ISBN 8021039051. info
  • HAMALA, Milan. Nelineárne programovanie. 2., dopl. vyd. Bratislava: Alfa, vydavateľstvo technickej a ekonomickej literatúry, 1976, 240 s. info
  • BERTSEKAS, Dimitri P. Convex Optimization Theory. Athena Scientific, 2009, 256 pp. ISBN 978-1-886529-31-1. info
  • Convex analysis. Edited by R. Tyrrell Rockafellar. Princeton: Princeton University Press, 1970, xviii, 451. ISBN 0691080690. info
  • BORWEIN, Jonathan M. and Adrian S. LEWIS. Convex analysis and nonlinear optimization : theory and examples. New York: Springer-Verlag, 2000, x, 273. ISBN 0387989404. info
  • SUN, Wenyu and Ya-Xiang YUAN. Optimization Theory and Methods - Nonlinear Programming. New York: Springer, 2006, 687 pp. Springer Optimization and Its Applications, Vol. 1. ISBN 978-0-387-24975-9. info
  • SUCHAREV, Aleksej Grigor‘jevič, Aleksandr Vasil'jevič TIMOCHOV and Vjačeslav Vasil'jevič FEDOROV. Kurs metodov optimizacii. Moskva: Nauka, 1986, 325 s. info
Teaching methods
Lectures and exercises.
Assessment methods
In order to be admitted to the exam, a semester project is required - the details are available in the study materials in IS. The standard lecture and seminar, the exam has both written and oral parts.

The conditions (especially regarding the form of the exam) will be specified according to the epidemiological situation and valid restrictions.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2010 - only for the accreditation, Autumn 2000, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Autumn 2022, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2022/M5170