Course detail

Optimization I

FSI-SOPAcad. year: 2018/2019

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.

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.


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


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Dupačová et al.: Lineárne programovanie, Alfa 1990 (CS)
Klapka a kol.: Metody operačního výzkumu, , 2001 (CS)
Dvořák a kol.: Operační analýza, , 1997 (CS)
Bazaraa et al.: Linear Programming and Network Flows, , Wiley 2011 (EN)
Bazaraa et al.: Nonlinear Programming, , Wiley 2012 (EN)
Charamza a kol.: Modelovací systém GAMS, , 1994 (EN)
Klapka a kol.: Metody operačního výzkumu, VUT 2001 (CS)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Graded course-unit credit is awarded based on the result in a written exam involving modelling-related, computational-based, and theoretical questions. The short oral exam is also included.

Language of instruction


Work placements

Not applicable.


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.

Classification of course in study plans

  • Programme B3A-P Bachelor's

    branch B-MAI , 3. year of study, summer semester, 4 credits, compulsory

Type of course unit



26 hours, optionally

Teacher / Lecturer


1. Introductory optimization: problem formulation and analysis, model building, theory.
2. Visualisation, algorithms, software, postoptimization.
3. Linear programming (LP): Convex and polyhedral sets.
4. LP: Feasible sets and related theory.
5. LP: The simplex method.
6. LP: Duality, sensitivity and parametric analysis.
7. Network flows modelling.
8. Introduction to integer programming.
9. Nonlinear programming (NLP): Convex functions and their properties.
10. NLP: Unconstrained optimization and line search algorithms.
11. NLP: Unconstrained optimization and related multivariate methods.
12. NLP: Constrained optimization and KKT conditions.
13. NLP: Constrained optimization and related multivariate methods.
14. Selected general cases.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer


Introductory problems (1-2)
Linear problems (2-7)
Special problems (7-8)
Nonlinear problems (9-13)
General problems (14)

Course participance is obligatory.