Course detail

Optimization

FIT-OPMAcad. year: 2024/2025

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)

Entry knowledge

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

Rules for evaluation and completion of the course

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

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.
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.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

Bazaraa et al.: Linear Programming and Network Flows, Wiley 1990.
Bazaraa et al.: Nonlinear Programming, Wiley 1993.
Charamza a kol.: Modelovací systém GAMS, Praha 1994.
Klapka a kol.: Metody operačního výzkumu, Brno 2001.

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., compulsory

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