Course Syllabus for

# Optimization Optimering

## FMAN60F, 6 credits

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

## General Information

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

## Aim

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.

## Goals

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.

## Course Contents

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.

## Course Literature

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

## Instruction Details

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

## Examination Details

Examination formats: Written exam, written assignments. Programming exercise with written report.
Examiner: