Optimering

**Valid from:** Autumn 2021**Decided by:** Professor Thomas Johansson**Date of establishment:** 2021-09-07

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course codes:** FMAN61, MATC61**Teaching language:** English

In many applications of mathematics, e.g. image analysis, control theory and time series analysis, an essential step is to choose the parameters in a model so that it fits given data as well as possible. One wants to minimize the error, measured in some way, which may be considered as a function of several variables – the parameters – that may have to satisfy further conditions – constraints. The aim of the course is to make the doctoral student familiar with the most common methods for solving optimization problems in which the parameters may vary continuously.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- in his/her own words, be able to describe the optimization algorithms, for problems with and without constraints, encountered in the course, and their properties.
- be able to outline those the theory of convex sets and convex functions that is included in the course, and be able to state and derive the most important theorems on convexity.
- be able to give examples of how to to make use of convexity in the treatment of an optimization problem.
- be able to outline the Kuhn-Tucker Theory and be able to state and derive the most important theorems therein.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to demonstrate an ability to solve optimization problems within the framework of the course.
- be able to demonstrate an ability to handle optimization problems using a computer.
- be able to demonstrate an ability to, in the context of problem solving, develop the theory somewhat further.
- be able to describe the connections between different concepts in the course, with proper terminology and in a well structured and logically consistent manner,.
- with proper terminology, suitable notation, in a well structured way and with clear logic be able to describe the solution to a mathematical problem and the theory within the framework of the course.

Quadratic forms and matrix factorisation. Convexity. Separatting planes and Farkas' lemma. The theory of optimization with and without constraints: Lagrange functions, Kuhn-Tucker theory. Duality. Methods for optimization without constraints: line search, steepest descent, Newton methods, conjugate directions, non-linear least squares optimization. The Nelder-Mead search method without derivatives. Methods for optimization with constraints: linear optimization, the simplex method, quadratic programming, penalty and barrier methods.

Böiers, L.: Mathematical Methods of Optimization. 2010. ISBN 9789144070759.

**Types of instruction:** Lectures, seminars, laboratory exercises, exercises

**Examination formats:** Written exam, written report.
Programming exercise with written report. **Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Calculus and linear algebra. Sufficient background is provided, e.g., by the courses FMAA05, FMA430, and FMAF05 or FMAF10.

Replaces FMAN60F.

**Course coordinators:** **Web page:** http://www.maths.lth.se/course/bigopt/