Course detail

Statistics and Optimization

FSI-USO-AAcad. year: 2024/2025

The course makes students familiar with introduction to operations research techniques for engineering problems. In the first part basic of probability theory and main principles of mathematical statistics (descriptive statistics, parameters estimation, tests of hypotheses, and linear regression analysis] are presented. The second part of the course deals with fundamental optimization models and methods for solving of technical problems. The principal ideas of mathematical programming are discussed: problem analysis, model building, solution search, especially and the interpretation of results. The particular results on linear and nonlinear programming are under focus.

Language of instruction

English

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

RNDr. Pavel Popela, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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

Rules for evaluation and completion of the course

Course-unit credit requirements: active participation in seminars, mastering the subject.The exam result 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.


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 make students familiar with basic concepts, methods and techniques of probability theory and mathematical statistics as well as with the development of stochastic way of thinking for modelling a real phenomenon and processes in engineering branches. The course objective is to also 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 or real-world data experience.


Students obtain the needed knowledge of the probability theory, descriptive statistics and mathematical statistics, which will enable them to understand and apply stochastic models of technical phenomena based upon these methods. Students will learn fundamental optimization topics (especially linear and non-linear programming). They will also become familiar with useful algorithms and interesting applications.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Bazaraa M. et al.: Linear Programming and Network Flows,. John Wiley and Sons, 2011 (EN)
Bazaraa, M. et al.: Nonlinear Programming,, John Wiley and Sons, 2012 (EN)
Boyd, S. and Vandeberghe, L.: Convex Optimization. Cambridge: Cambridge University Press, 2004. (EN)
Hahn, G. J. - Shapiro, S. S.: Statistical Models in Engineering.New York : John Wiley & Sons, 1994. (EN)
Montgomery, D. C. - Renger, G.: Applied Statistics and Probability for Engineers. New York : John Wiley & Sons, 2003. (EN)

Recommended reading

Karpisek, Z. Apllied Statistics, Brno, 2007
Montgomery, D. C. - Renger, G.: Applied Statistics and Probability for Engineers. New York : John Wiley & Sons, 2003.
Rardin, R. L. Optimization in Operations Research. Pearson, 2015.

Classification of course in study plans

  • Programme N-ENG-A Master's 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Pavel Popela, Ph.D.

Syllabus

1. Random events and their probability.
2. Random variable and vector, types, functional a numerical characteristics.
3. Basic discrete and continuous probability distributions.
4. Random sample, sample characteristics, and parameters estimation (point and interval estimates).
5. Testing statistical hypotheses
6. Introduction to regression analysis.
7. Introductory optimization: problem formulation and analysis, model building, theory.
8. Visualisation, algorithms, software, postoptimization.
9. Linear programming (LP): Convex and polyhedral sets. Feasible sets and related theory.
10. LP: The simplex method.
11. Nonlinear programming (NLP): Convex functions and their properties. Unconstrained optimization and selected algorithms.
12. NLP: Constrained optimization and KKT conditions.
13. NLP: Constrained optimization and related multivariate methods.

Exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Pavel Popela, Ph.D.

Syllabus

1. Descriptive statistics - examples.
2. Probability - basic examples.
3. Functional and numerical characteristics of random variable.
4. Selected probability distributions - examples.
5. Point and interval estimates of parameters - examples.
6. Testing hypotheses - examples.
7. Linear regression (straight line), estimates, tests and plots.
8. Introductory problems - formulation, model building.
9. Linear problems: extreme points and directions.
10. Linear problems: simplex method.
11. Nonlinear problems - examples of the algorithm use (unconstrained optimization) .
12. Nonlinear problems - KKT.
13. Nonlinear problems examples of the algorithm use (constrained optimization)