M5170 Mathematical Programming

Faculty of Science
Autumn 2016
Extent and Intensity
2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
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 19. 9. to Sun 18. 12. Wed 8:00–9:50 M1,01017
  • Timetable of Seminar Groups:
M5170/01: Mon 19. 9. to Sun 18. 12. Fri 11:00–11:50 M2,01021, P. Zemánek
M5170/02: Mon 19. 9. to Sun 18. 12. Fri 12:00–12:50 M1,01017, 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
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
    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
Theoretical lecture and seminar
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.
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 2017, Autumn 2018, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Autumn 2016, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2016/M5170