CTU FEE Moodle

The course provides an introduction to mathematical optimization, specifically to optimization in real vector spaces of finite dimension. The theory is illustrated with a number of examples. You will refresh and extend many topics that you know from linear algebra and calculus courses.

The aim of the course is to teach students to recognize optimization problems around them, formulate them mathematically, estimate their level of difficulty, and solve easier problems.

1. General problem of continuous optimization.
2. Matrices, linear and affine subspaces, orthogonality.
3. Overdetermined linear systems, least squares.
4. Quadratic forms and functions, definitness of a matrix, spectral decomposition.
5. Singular value decomposition (SVD), application in optimization.
6. Analytical conditions on unconstrained local optima.
7. Iterative methods for unconstrained local optima.
8. Local optima constrained by equalities, Lagrange multipliers.
9. Linear programming - intro.
10. Linear programming - applications.
11. Convex sets and polyhedra.
12. Linear programming - duality.
13. Convex functions.
14. Intro to convex optimization.

At seminars, students exercise the theory by solving problems together using blackboard and solve optimization problems in Matlab as homeworks.

Basic:
Online lecture notes Tomáš Werner: Optimalizace (see www pages of the course).

Optionally, selected parts from the books:
Lieven Vandenberghe, Stephen P. Boyd: Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares, Cambridge University Press, 2018.
Stephen Boyd and Lieven Vandenberghe: Convex Optimization, Cambridge University Press, 2004. (selected parts)

Linear algebra. Calculus, including intro to multivariate calculus. Recommended are numerical algorithms and probability and statistics.