Course detail

Optimization Models I

FSI-SOM-AAcad. year: 2025/2026

The course focuses on fundamental optimization models and methods for solving of logistic problems. The principal ideas of mathematical programming are presented in the following steps: 1) formulation and analysis of problems, 2) building of mathematical models, 3) classification of models, possible transformations and assessment of their theoretical properties, 4) selection and modification of solution algorithms, 5) their software implementation, 6) finding, analysis and interpretation of the optimal solutions. The course mainly covers the topics of linear programming (convex and polyhedral sets, simplex method, duality) and nonlinear programming (convex functions, optimality conditions, selected algorithms). The content of the
course was developed by the author on the basis of his experience with similar courses collected during his stays at foreign universities.

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

Basic concepts of calculus, linear algebra, and programming.

Rules for evaluation and completion of the course

Credit is awarded on the basis of the student's active participation in the course and his/her significant contribution to group homework projects during the semester. Then, the examination result is based on the results of a written paper including modeling-related, computational and theoretical questions. The written work is then accompanied by an oral discussion with the student. The attendance at seminars is required as well as active participation in lectures. Passive or missing students are required to work out additional
assignments.

Aims

The course focuses on introducing students to the fundamental knowledge of optimization - mathematical programming. Emphasis is placed on presenting essential information about models and methods for solving optimization problems. This includes an introduction to the analysis of decision problems, the building of basic (linear and nonlinear) mathematical models, their formal descriptions and analysis of their properties,
appropriate transformations of models in terms of their solvability, and the choice and modification of algorithms. The theoretical interpretation of the above mentioned knowledge is supported by illustrative geometrical and numerical examples with the aim to lead students to a deep and applicationally useful understanding of the lectured material. 

The course is designed for Logistics Analytics students and is useful for students of applied science and selected engineering disciplines. Participating students will gain knowledge of the theoretical foundations of optimization (especially linear and nonlinear programming). They will also learn general principles of modeling and selected algorithms for solving optimization problems, and form a basic idea of the use of optimization
models in typical applications.

Study aids

Prerequisites and corequisites

Not applicable.

Basic literature

Bazaraa et al.: Linear Programming and Network Flows, , Wiley 2011 (EN)
Bazaraa et al.: Nonlinear Programming , Wiley 2012 (EN)
Williams, H.P. Model Building in Mathematical Programming, 5th edition. J.Wiley and Sons, 2013. (EN)

Recommended reading

Bisschop, J. et al. AIMMS Optimization modeling, AIMMS Netherlands, 2023. (EN)
Boyd, S. and Vandeberghe, L.: Convex Optimization. Cambridge: Cambridge University Press, 2004. (EN)
Bynum, M.L. et al. Pyomo — Optimization Modeling in Python, 3rd edition, Springer  2021. (EN)
Rardin, R. L. Optimization in Operations Research. Pearson, 2015. (EN)

Classification of course in study plans

  • Programme N-LAN-A Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

RNDr. Pavel Popela, Ph.D.

Syllabus

1. Applications of principles of optimization model building (LP, NLP examples)
2. Basic optimization concepts illustrated by examples 
3. Principal algoithmic ideas by examples, graphical solution examples,  optimization software in use 
4. Basic theoretical concepts by examples for polyhedal sets, LP standard form, extreme points, extreme directions, and representation theorem
5. Key ideas of simplex method by examples, the use of optimality conditions, geometry examples, algebraic examples
6. Simplex method in use by tabular form and matrix form, Examples of two-phase and big M methods
7. Advanced topics of simplex method by examples including  degeneracy, convergence, cycling, and reduced costs
8. Syntax rules for duality in LP by examples. Examples of dual simplex method
9. Duality in LP semantics (weak duality, strong duality theorem, complementarity, shadow prices)
10. Advanced topics of LP by examples for sensitivity, ranges, changes, and paametric analysis
11. NLP unconstrained cases by examples for quadratic and special cases, both convex and nonconvex. Optimality conditions computed, analytical and basic numerical examples, reformulations 
12. NLP constrained cases by examples for quadratic programming and special cases. Optimality conditions computed and numerical examples
13. NLP special cases in logistics by examples

Lecture

26 hod., optionally

Teacher / Lecturer

RNDr. Pavel Popela, Ph.D.

Syllabus

1. Principles of Optimization Models Building (LP, NLP examples, classification, steps and rules)
2. Basic Optimization Concepts (continuity, compactness, convexity - sets and functions)
3. Introduction to Principal Algoithmic Ideas (graphical solution, optimization related software overview)
4. Basic Theoretical Concepts (polyhedal sets, LP standard form and its properties, extreme points, extreme directions, representation theorem)
5. Key Ideas of Simplex Method (optimality conditions, geometry, algebraic approach, compact descriptions)
6. Simplex Method in Use (tabular form, matrix form, two-phase method, Big M method)
7. Advanced Topics of Simplex Method (degeneracy, convergence, cycling, memory and iteration requirements, reduced costs)
8. Duality in LP syntax rules (canonical and standard form, rules, dual simplex method)
9. Duality in LP semantics (weak duality, strong duality theorem, complementarity, shadow prices)
10. Advanced Topics of LP (sensitivity, ranges, changes, paametric analysis)
11. NLP unconstrained cases (quadratic and special cases, convex and nonconvex, optimality conditions, analytical solution, basic numerical approaches, reformulations)
12. NLP constrained cases (quadratic programming formulation and special cases, optimality conditions - KKT, remark on numerical techniques)
13. NLP special cases in logistics