|Vyučováno v||Winter and summer|
The course provides the basics of mathematical optimization: using linear algebra for optimization (least squares, SVD), Lagrange multipliers, selected numerical algorithms (gradient, Newton, Gauss-Newton, Levenberg-Marquardt methods), linear programming, convex sets and functions, intro to convex optimization, duality.
Linear algebra. Calculus, including intro to multivariate calculus. Recommended are numerical algorithms and probability and statistics.
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. Over-determined linear systems, method of least squares.
3. Minimization of quadratic functions.
4. Using SVD in optimization.
5. Algorithms for free local extrema (gradient, Newton, Gauss-Newton, Levenberg-Marquardt methods).
6. Linear programming.
7. Simplex method.
8. Convex sets and polyhedra. Convex functions.
9. Intro to convex optimization.
10. Lagrange formalism, KKT conditions.
11. Lagrange duality. Duality in linear programming.
12. Examples of non-convex problems.
13. Intro to multicriteria optimization.
At seminars, students exercise the theory by solving problems together using blackboard and solve optimization problems in Matlab as homeworks.
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.