Course detail

# Optimization

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.

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.

Recommended optional programme components

Not applicable.

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

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

The attendance at seminars is required as well as active participation. Passive or missing students are required to work out additional assignments.
Exam prerequisites:
Gaining at least 20 points during the semester.

Language of instruction

Czech

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.

Classification of course in study plans

• Programme IT-MGR-2 Master's

branch MBI , any year of study, summer semester, 4 credits, elective
branch MPV , any year of study, summer semester, 4 credits, elective
branch MSK , any year of study, summer semester, 4 credits, elective
branch MBS , any year of study, summer semester, 4 credits, elective
branch MMI , any year of study, summer semester, 4 credits, elective
branch MMM , any year of study, summer semester, 4 credits, compulsory-optional

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

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 hours, compulsory

Teacher / Lecturer