Course detail

Optimization

FIT-OPMAcad. year: 2017/2018

The course presents fundamental optimization models and methods for solving of technical problems. The principal ideas of mathematical programming are discussed: problem analysis, model building, solution search, and the interpretation of results. The course mainly deals with linear programming (polyhedral sets, simplex method, duality) and nonlinear programming (convex analysis, Karush-Kuhn-Tucker conditions, selected algorithms). Basic information about network flows and integer programming is included as well as further generalizations of studied mathematical programs.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

RNDr. Pavel Popela, Ph.D.

Department

Dept. of Statistics and Optimization (ÚM OSO)

Learning outcomes of the course unit

The course is designed for mathematical engineers and it is useful for applied sciences students. Students will learn the theoretical background of fundamental topics in optimization (especially linear and non-linear programming). They will also made familiar with useful algorithms and interesting applications.

Prerequisites

Fundamental knowledge of principal concepts of Calculus and Linear Algebra in the scope of the mathematical engineering curriculum is assumed.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Gaining at least 20 points during the semester.

Course curriculum

    Syllabus of lectures:
    1. Introductory models (IM): problem formulation, problem analysis, model design, theoretical properties.
    2. IM: visualization, algorithms, software, postprocessing in optimization
    3. Linear programming (LP): Convex and polyhedral sets.
    4. LP: Set of  feasible solutions and theoretical foundations.
    5. LP: The Simplex method.
    6. LP: Duality and parametric analysis.
    7. Network flow models.
    8. Basic concepts of integer programming.
    9. Nonlinear programming (NLP): Convex functions and their properties.
    10. NLP: Unconstrained optimization. Numerical methods for univariate optimization.
    11. NLP: Unconstrained optimization and related numerical methods for multivariate optimization.
    12. NLP: Constrained optimization and Karush-Kuhn-Tucker conditions.
    13. NLP: Constrained optimization and related numerical methods for multivariate optimization.

Work placements

Not applicable.

Aims

The course objective is to emphasize optimization modelling together with solution methods. It involves problem analysis, model building, model description and transformation, and the choice of the algorithm. Introduced methods are based on the theory and illustrated by geometrical point of view.

Specification of controlled education, way of implementation and compensation for absences

The attendance at seminars is required as well as active participation. Passive or missing students are required to work out additional assignments.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Dupačová et al.: Lineárne programovanie, Alfa, 1990. Bazaraa et al.: Linear Programming and Network Flows, Wiley 1990. Bazaraa et al.: Nonlinear Programming, Wiley 1993.

Recommended reading

Klapka a kol.: Metody operačního výzkumu, Brno 2001. Dvořák a kol.: Operační analýza, Brno, 1996. Charamza a kol.: Modelovací systém GAMS, Praha 1994. Dupačová et al.: Lineárne programovanie, Alfa, 1990. Bazaraa et al.: Linear Programming and Network Flows, Wiley 1990. Bazaraa et al.: Nonlinear Programming, Wiley 1993.

Classification of course in study plans

  • Programme IT-MSC-2 Master's

    branch MMI , 0 year of study, summer semester, elective
    branch MBI , 0 year of study, summer semester, elective
    branch MSK , 0 year of study, summer semester, elective
    branch MMM , 0 year of study, summer semester, compulsory-optional
    branch MBS , 0 year of study, summer semester, elective
    branch MPV , 0 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Pavel Popela, Ph.D.

Syllabus

  1. Introductory models (IM): problem formulation, problem analysis, model design, theoretical properties.
  2. IM: visualization, algorithms, software, postprocessing in optimization
  3. Linear programming (LP): Convex and polyhedral sets.
  4. LP: Set of  feasible solutions and theoretical foundations.
  5. LP: The Simplex method.
  6. LP: Duality and parametric analysis.
  7. Network flow models.
  8. Basic concepts of integer programming.
  9. Nonlinear programming (NLP): Convex functions and their properties.
  10. NLP: Unconstrained optimization. Numerical methods for univariate optimization.
  11. NLP: Unconstrained optimization and related numerical methods for multivariate optimization.
  12. NLP: Constrained optimization and Karush-Kuhn-Tucker conditions.
  13. NLP: Constrained optimization and related numerical methods for multivariate optimization.

Exercise in computer lab

13 hod., optionally

Teacher / Lecturer

RNDr. Pavel Popela, Ph.D.

Syllabus

  • Cvičení 1-2: Úvodní úlohy
  • Cvičení 2-7: Lineární úlohy
  • Cvičení 7-8: Speciální úlohy
  • Cvičení 9-13: Nelineární úlohy