Course detail

Optimization Methods I

FSI-FOA-A Acad. year: 2027/2028 Summer semester

The course provides a systematic introduction to mathematical optimization and computational methods for solving optimization problems in engineering applications. It focuses on mathematical problem formulation, analysis of problem properties, and selection of suitable optimization algorithms with respect to problem structure, computational requirements, and available information.

The first part of the course covers one-dimensional and multidimensional continuous optimization, first- and second-order methods, and derivative-free direct and stochastic optimization methods. This is followed by population-based optimization algorithms, convex optimization, optimality conditions, duality, and interior-point methods. The course also introduces linear and integer programming, the simplex method, branch-and-bound algorithms, and cutting-plane methods.

The final part addresses multiobjective optimization, surrogate-assisted optimization of computationally expensive problems, and optimization under uncertainty.

The course combines the mathematical foundations of optimization methods with their implementation and experimental evaluation. Students solve selected optimization problems, compare the behavior of different algorithms, and assess their suitability for specific applications.

Language of instruction

English

Number of ECTS credits

5

Supervisor

doc. Ing. Jakub Kůdela, Ph.D.

Department

Institute of Automation and Computer Science

Entry knowledge

Students are expected to have a basic understanding of linear algebra, including vector and matrix operations, systems of linear equations, and fundamental matrix properties.

Required knowledge of mathematical analysis includes differential calculus of functions of one and several variables, including gradients, partial derivatives, and Hessian matrices. A basic understanding of probability theory and mathematical statistics is also expected.

Students should be able to formulate a simple mathematical problem, interpret its solution, and implement basic numerical algorithms in a suitable programming environment.

Rules for evaluation and completion of the course

Course-unit credit: Active participation in the seminars, elaboration of a given project. Examination: Written.
Attendance at seminars is controlled. An absence can be compensated for via solving additional problems.

Aims

Upon successful completion of the course, students will be able to:

  1. Formulate and classify mathematical optimization problems by identifying objective functions, decision variables, and constraints, and justify the selection of suitable optimization methods based on problem characteristics.

  2. Apply and analyze selected one-dimensional and multidimensional continuous optimization methods, including first- and second-order methods, and evaluate their convergence, accuracy, and computational requirements.

  3. Assess the convexity of optimization problems, apply fundamental optimality conditions and duality, and solve selected constrained optimization problems using interior-point methods, linear programming, and the simplex method.

  4. Formulate and solve selected integer and combinatorial optimization problems and explain the principles of branch-and-bound and cutting-plane methods.

  5. Implement and experimentally compare selected direct, stochastic, and population-based optimization algorithms, evaluating solution quality, computational cost, and sensitivity to algorithm parameters.

  6. Apply and evaluate selected approaches to multiobjective optimization, surrogate-assisted optimization, and optimization under uncertainty, and interpret their results in the context of the problem being solved.

Study aids

Students are provided with lecture presentations, exercise assignments, sample calculations, implementations of selected optimization algorithms, and test optimization problems. The materials are made available via the university's e-learning environment.

The study programmes with the given course

Programme N-AIŘ-P: Applied Computer Science and Control, Master's, compulsory

Type of course unit

 

Lecture

39 hours, optionally

Syllabus

1. Introduction to optimization.
2. 1D optimization methods.
3. First and second-order methods.
4. Direct methods and stochastic methods.
5. Population methods, metaheuristics.
6. Convexity theory, KKT conditions, duality.
7. Interior point methods.
8. Linear programming.
9. Simplex method.
10. Integer and combinatorial problems, Branch and bound method, Gomory cuts.
11. Multicriteria optimization.
12. Surrogate-assisted optimization.
13. Optimization under uncertainty.

Exercise

12 hours, compulsory

Syllabus

Implementation and analysis of the following methods:
1. – 2. Methods for 1D optimization.
3. First-order methods.
4. Second-order methods.
5. Direct methods.
6. Stochastic methods.
7. Population methods, metaheuristics.
8. Interior point methods.
9. Simplex method.
10. Branch and bound method, Gomory cuts.
11. Multicriteria optimization methods.
12. Surrogate-assisted methods.
13. Methods for uncertain data.

Computer-assisted exercise

14 hours, compulsory

Syllabus

Implementation and analysis of the following methods:
1. – 2. Methods for 1D optimization.
3. First-order methods.
4. Second-order methods.
5. Direct methods.
6. Stochastic methods.
7. Population methods, metaheuristics.
8. Interior point methods.
9. Simplex method.
10. Branch and bound method, Gomory cuts.
11. Multicriteria optimization methods.
12. Surrogate-assisted methods.
13. Methods for uncertain data.