Optimering

**Valid from:** Autumn 2019**Decided by:** FN1/Anders Gustafsson**Date of establishment:** 2014-01-27

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course codes:** FMAN60, FMA051**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

- be familiar with and, 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 familiar with the theory of convex sets and convex functions, and be able to state and derive the most important theorems on convexity.
- be aware of how to make use of convexity in the treatment of an optimization problem.
- be familiar with 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. 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. 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 assignments.
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 FMA051F.

**Course coordinators:**