Course details

Optimization

OPM Acad. year 2024/2025 Summer semester 4 credits

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.

Guarantor

Popela Pavel, RNDr., Ph.D. (IM DSO)

Language of instruction

Czech

Completion

Credit+Examination (written+oral)

Time span

  • 26 hrs lectures
  • 13 hrs pc labs

Assessment points

  • 60 pts final exam
  • 40 pts projects

Department

Dept. of Statistics and Optimization (ÚM OSO)

Learning objectives

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.

Prerequisite knowledge and skills

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

Study literature

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

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.

Syllabus of computer exercises

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

Progress assessment

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.

Back to top